Mappings between codewords of two distinct (N,K) Reed-Solomon codes over GF (2J)

ABSTRACT

A process for realizing mappings between codewords of two distinct (N,K) Reed-Solomon codes over GF(2 J ) having selected two independent parameters: J, specifying the number of bits per symbol; and E, the symbol error correction capability of the code, wherein said independent parameters J and E yield the following: N=2 J  -1, total number of symbols per codeword; 2E, the number of symbols assigned a role of check symbols; and K=N-2E, the number of code symbols representing information, all within a codeword of an (N,K) RS code over GF(2 J ), and having selected said parameters for encoding, the implementation of a decoder are governed by: 2 J  field elements defined by a degree J primitive polynomial over GF(2) denoted by F(x); a code generator polynomial of degree 2E containing 2E consecutive roots of a primitive element defined by F(x); and, in a Berlekamp RS code, the basis in which the RS information and check symbols are represented. 
     The process includes separate transformation steps for symbol-by-symbol conversion for a first RS code to ultimately a second conventional RS code capable of being corrected by a conventional RS decoder, followed by a reverse sequence of the inverse of the first set of steps to arrive at codewords having corrected information symbols, at which time check symbols of the RS code may be discarded.

ORIGIN OF INVENTION

The invention described herein was made in the performance of work under a NASA contract, and is subject to the provisions of Public Law 96-517 (35 USC 202) in which the contractor has elected not to retain title.

TECHNICAL FIELD

This invention relates to error-correcting codes, and more particularly to mappings between two distinct (N,K) Reed-Solomon (RS) codes over GF(2^(J)).

BACKGROUND ART

An RS codeword is composed of N J-bit symbols of which K represents information and the remaining N-K symbols represent check symbols. Each check symbol is a linear combination of a distant subset of information symbols. RS codes have the following parameters:

    ______________________________________                                         J            number of bits comprising each RS                                              symbol                                                            N = 2.sup.J - 1                                                                             number of symbols per RS codeword                                 E            symbol correction capability per                                               RS codeword                                                       2E           number of checks symbols                                          K = N - 2E   number of symbols representing                                                 information                                                       I            the depth of symbol interleaving                                  ______________________________________                                    

Parameters J, E and I are independent. Each of the 2E check symbols (computed by an RS encoder) is a linear combination of a distinct subset of the K symbols representing information. Hence (N,K) RS codes are linear. Furthermore, every cyclic permutation of the symbols of an RS codeword results in a codeword.

The class of cyclic codes is a proper subset of the class of linear codes. Cyclic codes have a well defined mathematical structure. Encoders and decoders of cyclic codes are implementable by means of feedback shift registers (FSRs). However, RS codes are nonbinary and each stage must be capable of assuming each of the 2^(J) state-values corresponding to representations of J-bit symbols. Consequently, solid-state random-access memories (RAMs) are commonly used to serve as nonbinary FSR stages in RS encoders and decoders.

Every pair of distinct codewords belonging to a (N,K) RS code differs in at least 2E+1 corresponding symbols. The code thus has a minimum Hamming distance of 2E+1 and is E symbol error-correcting. A symbol is in error if one or more bits comprising the symbol are in error. A received word with any combination of E or fewer symbols in error will be correctly decoded. Whereas a received word containing more than E erroneous symbols will be incorrectly decoded with a probability of less than 1 chance in E factorial (i.e., 1/E!). Such a received word will be declared to be uncorrectable during the decoding process with probability 1-(1/E!).

Erroneous symbols (of a received word) confined to a region of E or fewer contiguous symbols in bit serial form are correctable. In terms of bits, every single burst-error (contained within a received word) of length J(E-1)+1 bits or less affects at most E symbols and thus is correctable.

A concatenated (NI,KI) RS code resulting from symbol interleaving to a depth of I is comprised of KI consecutive information symbols over which 2EI check symbols are computed and appended such that every I^(th) symbol, starting with symbol 1,2, . . . or I, belongs to the same (N,K) RS codeword. If a received word of a (NI,KI) code contains any single burst-error of length J(EI-1)+1 bits, the number of erroneous symbols belonging to the same N-symbol word will not exceed E. Upon symbol deinterleaving, each of the I, N-symbol words will be correctly decoded. Thus symbol interleaving to a depth of I increases the length (in bits) of correctable burst-errors by over I-fold.

It is the burst-error correction capability of RS codes that is exploited in concatenated coding where a convolution (probabilistic) code is the inner code and an RS (algebraic) code serves as the outer code. A large number of bit-errors within a burst results in a relatively few number of symbol errors. Convolutional codes outperform algebraic codes over a Gaussian channel. Burst-errors are encountered in a low signal-to-noise ratio environment. Also, in such an environment, a Viterbi decoder (of a convolutional code) can lose bit or symbol synchronization. This results in the generation of bursts by the Viterbi decoder. Until synchronization is re-established, burst-error protection is provided by the outer RS code. Expectation of burst-lengths among other factors (such as information rate K/N, a measure of efficiency, and transfer frame length in packet telemetry) influence the choice of magnitudes of parameters E and I. The effect of an infinite symbol interleaving depth can be achieved with a small value of I.

Mathematical Characterization of RS Symbols

A codeword of a (N,K) RS code is represented by a vector of N symbols (i.e., components)

    C.sub.N-1 C.sub.N-2. . . C.sub.2E C.sub.2E-1. . . C.sub.0  ( 1)

where C_(i) is taken from a finite field 2^(J) elements. The finite field is known as a Galois Field of order 2^(J) or simply GF(2^(J)). Each element in GF(2^(J)) is a distinct root of

    x.sup.2.spsp.J -x=x(x.sup.2.spsp.J.sbsp.-1 -1)=0           (2)

The element O satisfies x=0 and each of the nonzero elements satisfies

    x.sup.2.spsp.J.sbsp.-1 -1=0                                (3)

The nonzero roots of unity in Equation (3) form a "multiplicative group" which is cyclic and of order N=2^(J) -1. The "multiplicative" order of a root, α, of Equation (3) is the least positive integer m for which

    α.sup.m =1 where 2.sup.J -1.tbd.0 mod m              (4)

Note that m|2^(J) -1 where u|v denotes u divides v. A root of maximum order (i.e., for which m=2^(J) -1) is termed "primitive" and is a "generator" of the cyclic multiplicative group in GF(2^(J)).

An irreducible (i.e., unfactorable) polynomial over a finite field is analogous to a prime integer. A fundamental property of Galois fields is that every irreducible polynomial F(x) of degree r over GF(2) where r|J (excluding x the only irreducible polynomial over GF(2) without a nonzero constant term) is a factor of

    x.sup.2.spsp.J.sbsp.-1 -1

Though F(x) is unfactorable over the binary field (i.e., GF(2)), it contains r distinct roots in GF(2^(J)). GF(2^(J)) is a finite field extension of GF(2), whereas GF(2) is a proper subfield of GF(2^(J)). GF(2^(s)) is a proper subfield of GF(2^(J)) if and only if s|J and s<J.

EXAMPLE 1

Given the irreducible polynomial of degree r=J=4

    F(x)=x.sup.4 +x.sup.3 +1 over GF(2)

Let α, among the 15 nonzero elements in GF(2⁴), denote a root of F(x). Then

    F(x)=α.sup.4 +α.sup.3 +1=0 and α.sup.4 =α.sup.3 +1

where "+" denotes "sum modulo 2" (i.e., the Exclusive-OR operation). Repeated multiplicative operations on α gives

    α, α.sup.2, α.sup.3, α.sup.4 +α.sup.3 +1, α.sup.5 =α.sup.4 +α=α.sup.3 +α+1, . . . and α.sup.15 =1=α.sup.0

Thus α, a root of F(x), is primitive and F(x) is a primitive polynomial over GF(2). The elements of GF(2⁴) are expressible as powers of α, each of which is equal to one of 16 distinct polynomials in α of degree less than 4 over GF(2) as follows:

    α.sup.i =b.sub.3 α.sup.3 +b.sub.2 α.sup.2 +b.sub.1 α+b.sub.0 where b.sub.j ε GF(2)

The element O is the constant zero polynomial denoted by

    α*=0.α.sup.3 +0.α.sup.2 +0.α+0

The elements of GF(2⁴) generated by α with α* adjoined appear in Table I.

                  TABLE I                                                          ______________________________________                                         GF(2.sup.4) Generated by α, a Root of F(x) =                             x.sup.4 + x.sup.3 + 1 over GF(2), with α*                                Adjoined.                                                                      i of α.sup.i                                                                          b.sub.3                                                                              b.sub.2     b.sub.1                                                                            b.sub.0                                     ______________________________________                                         *            0     0           0   0                                           0            0     0           0   1                                           1            0     0           1   0                                           2            0     1           0   0                                           3            1     0           0   0                                           4            1     0           0   1                                           5            1     0           1   1                                           6            1     1           1   1                                           7            0     1           1   1                                           8            1     1           1   0                                           9            0     1           0   1                                           10           1     0           1   0                                           11           1     1           0   1                                           12           0     0           1   1                                           13           0     1           1   0                                           14           1     1           0   0                                           ______________________________________                                    

The binary operation of "addition" is termwise sum modulo 2 (or equivalently, vector addition over GF[2]).

    α.sup.5 +α.sup.13 =[1011]+[0111]=[1101]=α.sup.11

The binary operation of "multiplication" is polynomial multiplication subject to the rules of modulo 2 arithmetic.

    (b.sub.3 α.sup.3 +b.sub.2 α.sup.2 +b.sub.1 α+b.sub.0) (d.sub.3 α.sup.3 +d.sub.2 α.sup.2 +d.sub.1 α+d.sub.0),

with the result reduced modulo α⁴ +α³ +1. Since each element is expressible as a power of α, "multiplication" is simplified with

    (α.sup.i) (α.sup.j)=α.sup.i+j mod 15

The logarithm to the base α for each field element appears in Table I. Thus

    (α.sup.14) (α.sup.6)=α.sup.5 =[1011]

Note that a more descriptive but less convenient representation of the O element is α⁻∞.

    α*α.sup.j =α* and α*+α.sup.j =α.sup.j for all j

Clearly commutativity holds for all field elements under both binary operations (a property of fields).

Consider the operation σ which squares each of the roots of ##EQU1## any polynomial of degree r over GF(2). Solomon W. Golomb, "Theory of Transformation Groups of Polynomials over GF(2) with Applications to Linear Shift Register Sequences," American Elsevier Publishing Company, Inc., 1968, pp. 87-109. Since ##EQU2## The substitution t² =x is appropriately employed in proving that f(x) over GF(2) is invariant under the root-squaring operation σ. The operation σ on f(x) which leaves f(x) unchanged is termed an automorphism. An operation on a root of f(x) is an automorphism if and only if it is an integer power of σ, the root-squaring operation. If, in particular, f(x) of degree r is irreducible over GF(2), then f(x) has the following r distinct automorphisms:

    1, α, α.sup.2, . . . , α.sup.r-1

with respect to GF(2^(r)). Consequently, f(x) has r distinct roots, namely,

    α, α.sup.2, α.sup.2.spsp.2, . . . , α.sup.2.spsp.r-1

Since σ^(r) maps α into α².spsp.r and α².spsp.r =α (from α².spsp.r.sbsp.-1 =1), σ^(r) is the identity operation.

Consider the element β among the 2^(r) -1 nonzero elements in GF(2^(r)). If β has order m, then β^(j) has order m/(m,j), where (m,j) denotes the Greatest Common Divisor (gcd) of m and j. Clearly, m|(2^(r) -1) and (m/(m,j))|(2^(r) -1). Recall that a primitive polynomial of degree r has α, a primitive root of unity, as a root. Each of the r roots has order 2^(r) -1 and is thus primitive since (2^(r) -1,2^(i))=1 for all i.

The set of integers

    {i}={1, 2, 2.sup.2, . . . , 2.sup.r-1 }

taken from the multiplicative group of integers modulo 2^(r) -1 form a subgroup. The corresponding set

    {α.sup.i }={α, α.sup.2, α.sup.2.spsp.2, . . . , α.sup.2.spsp.r-1 }

are the r roots of a primitive r^(th) degree polynomial over GF(2). The "generalized cosets"

    {v, 2v, 2.sup.2 v, . . . , (2.sup.r-1)v}

are nonoverlapping sets which together with the subgroup {i}, the special coset where v=1, comprise the multiplicative group of integers modulo 2^(r) -1. A one-to-one correspondence exists between the elements of the group and the 2^(r) -1 roots of unity contained in GF(2^(r)). If (2^(r) -1, v)=1, then {iv} is a coset as defined in group theory. The elements of such a coset correspond to r (2^(r) -1)^(st) primitive roots whose minimal polynomial is an r^(th) degree primitive polynomial over GF(2). The polynomial of least degree that contains a given set of distinct roots (corresponding to a coset) is irreducible and termed a minimal polynomial.

The number of positive integers no greater than n (a positive integer) that are relatively prime to n is the number-theoretic function φ(n) known as Euler's phi function. Two integers n and i are termed relatively prime if (n, i)=1. Given α, a primitive root of unity in GF(2^(r)), then α^(j) is primitive if and only if (2^(r) -1, j)=1. There are a total of φ(2^(r) -1) primitive roots falling into (φ(2^(r) -1))/r groupings corresponding to cosets. The r roots in each grouping are roots of a distinct primitive polynomial and thus there are (φ(2^(r) -1))/r primitive polynomials of degree r over GF(2).

An "improper coset" results for values of v where (2^(r) -1)≠1. If the coset contains r distinct elements, the elements correspond to r (2^(r) -1)^(st) nonprimitive roots of unity whose minimal polynomial is an irreducible nonprimitive r^(th) degree polynomial over GF(2). Whereas the elements of a coset containing s<r distinct elements (where s necessarily divides r) correspond to s (2^(s) -1)^(st) roots of unity whose minimal polynomial is an irreducible polynomial of degree s over GF(2).

Complete factorization of x².spsp.4⁻¹ for r = J = 4 appears in Table II.

                  TABLE II                                                         ______________________________________                                         Irreducible Factors of x.sup.2.spsp.4.sup.-1 - 1 for r = J = 4 over            GF(2).                                                                                                             order of                                   Cosets  Roots       Polynomial      Roots                                      ______________________________________                                         0       α.sup.0                                                                              x + 1           1                                          1  2  4 α, α.sup.2, α.sup.4, α.sup.8                                       x.sup.4 + x.sup.3 + 1                                                                          15                                         3  6 12 α.sup.3, α.sup.6, α.sup.12, α.sup.9                                x.sup.4 + x.sup.3 + x.sup.2 + x + 1                                                            5                                          5 10    α.sup.5, α.sup.10,                                                             x.sup.2 + x + 1 3                                          7 14 13 11                                                                             α.sup.7, α.sup.14, α.sup.13, α.sup.11                              x.sup.4 + x + 1 15                                         ______________________________________                                    

A degree 4 primitive polynomial is required to derive Tables I and II. Every irreducible polynomial over GF(2) of degree r≦16 can be determined from Tables in Appendix C of W. W. Peterson and Error-Correcting Codes, MIT Press, Cambridge, Mass., 1961. There are two primitive polynomials over GF(2) (which may be verified by the enumeration given by the number-theoretic function involving Euler's phi function). As shown in Table II, these are

    F.sub.B (x)=x.sup.4 +x.sup.3 +1 and F.sub.G (x)=x.sup.4 +x+1

The nonzero elements in Table I were generated by α, a root of F_(B) (x), and used in determining the irreducible polynomial factors in Table II. Irreducibility is a necessary but not sufficient condition for a polynomial over GF(2) to be primitive. Note that the roots of F_(G) (x) are reciprocals of the respective roots of F_(B) (x). That is

    α.sup.-1 =α.sup.15-1 =α.sup.14,

    α.sup.-2 =α.sup.13,

    α.sup.-4 =α.sup.11, and

    α.sup.-8 =α.sup.7,

and F_(B) (x) and F_(G) (x) are reciprocal polynomials over GF(2) where

    x.sup.4 F.sub.B (1/x)=F.sub.G (x)

The reciprocal of a primitive polynomial is a primitive polynomial.

The (nonprimitive) irreducible polynomial with roots α³, α⁶, α¹² and α⁹ is determined by using entries in Table I and appropriate field operations to expand

    (x-α.sup.3) (x-α.sup.6) (x-α.sup.12) (x-α.sup.9)

or by determining the b_(i) 's ε GF(2) which satisfy

    α.sup.12 +b.sub.3 α.sup.9 +b.sub.2 α.sup.6 +b.sub.1 α.sup.3 +1=0

The coefficients are linearly dependent vectors (i.e., polynomial coefficient strings). The latter method is easily programmable for computing the minimal polynomial (primitive or irreducible but nonprimitive) of degree r over GF(2) containing any given set of r distinct roots.

The elements α* , α⁰, α⁵ and α¹⁰ are roots of

    x.sup.4 -x=x(x.sup.3 -1)=x(x+1) (x.sup.2 +x+1) over GF(2)

and members of the subfield GF(2²)<GF(2⁴). Whereas the elements α* and α⁰ are roots of

    x.sup.2 -x=x(x+1) over GF(2)

and members of the subfield GF(2)<GF(2²)<GF(2⁴).

In Table I, GF(2⁴) is defined by the primitive polynomial

    F.sub.B (x)=x.sup.4 +x.sup.3 +1                            (5)

The 15 nonzero elements can also be generated by β, a root of the primitive polynomial

    F.sub.G (x)=x.sup.4 +x+1                                   (6)

and by adjoining β* another field of order 2⁴ results. All finite fields of the same order, however, are isomorphic. Two fields with different representations are said to be isomorphic if there is a one-to-one onto mapping between the two which preserves the operations of addition and multiplication. The one-to-one mapping of the field elements defined by F_(B) (x) in Equation (5) onto the field elements defined by F_(G) (x) in Equation (6) appears in Table III below.

                  TABLE III                                                        ______________________________________                                         A One-to-One Mapping of Elements of GF(2.sup.4)                                Defined by F.sub.B (x) = x.sup.4 + x.sup.3 + 1 Onto Elements                   of GF(2.sup.4) Defined by F.sub.G (x) = x.sup.4 + x + 1                        i of                           7i mod 15                                       a.sup.i                                                                               b.sub.3                                                                              b.sub.2                                                                              b.sub.1                                                                            b.sub.0 of β.sup.7i                                                                       c.sub.3                                                                            c.sub.2                                                                            c.sub.1                                                                            c.sub.0                     ______________________________________                                         *      0     0     0   0       *       0   0   0   0                           0      0     0     0   1   ⃡                                                                      0       0   0   0   1                           1      0     0     1   0   →                                                                           7       1   0   1   1                           2      0     1     0   0   →                                                                           14      1   0   0   1                           3      1     0     0   0   →                                                                           6       1   1   0   0                           4      1     0     0   1       13      1   1   0   1                           5      1     0     1   1       5       0   1   1   0                           6      1     1     1   1       12      1   1   1   1                           7      0     1     1   1       4       0   0   1   1                           8      1     1     1   0       11      1   1   1   0                           9      0     1     0   1   ←                                                                             3       1   0   0   0                           10     1     0     1   0       10      0   1   1   1                           11     1     1     0   1   ←                                                                             2       0   1   0   0                           12     0     0     1   1       9       1   0   1   0                           13     0     1     1   0   ←                                                                             1       0   0   1   0                           14     1     1     0   0       8       0   1   0   1                           ______________________________________                                    

The mappings α^(i) →β^(7i) and β^(7i) →α^(i) can be done by table look-up. They can also be realized by linear transformations. Consider the nonsingular matrix ##EQU3## Refer to Table III. Each row vector is defined by F_(G) (x) (having β as a root) and corresponds to a distinct unit vector defined by F_(B) (x) (having α as a root). The four unit vectors are a natural basis for elements [b₃ b₂ b₁ b₀ ], coefficient strings of polynomials in α. Thus

    [b.sub.3 b.sub.2 b.sub.1 b.sub.0 ]M.sub.αβ =[c.sub.3 c.sub.2 c.sub.1 c.sub.0 ]

is a linear transformation that maps α^(i) →β^(7i). Similarly ##EQU4## is a linear transformation that maps β^(7i) →α^(i). Clearly, M.sub.αβ⁻¹ =M.sub.βα.

The roots α and β⁷ both have minimal polynomial

    x.sup.4 +x.sup.3 +1

and α^(i) ⃡β^(7i) is one isomorphism between the two representations of GF(2⁴). Note that corresponding elements have the same multiplicative order. (Another isomorphism is α^(7i) ⃡β^(i) where α⁷ and β both have minimal polynomial x⁴ +x+1). Example 2 illustrates preservation of field operations.

EXAMPLE 2

Refer to entries in Table II.

    __________________________________________________________________________          1  0 0 1 - α.sup.4                                                                         ⃡                                                                    β.sup.13                                                                       = 1 1 0 1                                           + 0  1 0 1 = α.sup.9                                                                         ⃡                                                                    β.sup.3                                                                        = 1 0 0 0                                           = 1  1 0 0 = α.sup.14                                                                        ⃡                                                                    β.sup.8                                                                        = 0 1 0 1                                        and                                                                                 α.sup.10                                                                        = 1 0 1  0 ⃡                                                                    0  1 1 1 = β.sup.10                                 ×                                                                          α.sup.8                                                                         = 1 1 1  0 ⃡                                                                    1  1 1 0 = β.sup.11                                 = α.sup.3                                                                         = 1 0 0  0 ⃡                                                                    1  1 0 0 = β.sup.6                               __________________________________________________________________________

Berlekemp's Representation of RS Symbols Using the Concept of a Trace

Consider the field elements

    α.sup.i =b.sub.3 α.sup.3 +b.sub.2 α.sup.2 +b.sub.1 α+b.sub.0 for i=*, 0, . . . , 14                    (7)

in GF(2₄) defined by F_(B) (x) in Equation (5). See Tables II and III. The trace of element α is defined as ##EQU5## The trace of roots of unity corresponding to members of the same coset in Table II are identical. For example, ##EQU6##

In general the trace Tr is a function on GF(p^(r)) over GF(p) where p is a prime. Of particular interest is GF(2^(r)) over GF(2) where Tr is defined by ##EQU7##

The trace has the following properties:

(1) Tr(γ) ε GF(2)

(2) Tr(γ+δ)=Tr(γ)+Tr(δ)

(3) Tr(cγ)=cTr(γ) where ε GF(2)

(4) Tr(1)=r mod 2

Property (1) follows from ##EQU8## which implies that Tr(γ) ε GF(2). Proofs of properties (1), (2) and (3) for traces of elements in GF(p^(r)) appear in M. Perlman and J. Lee, Reed-Solomon Encoders Conventional vs Berlekamp's Architecture," JPL Publication 82-71, Dec. 1, 1982. Property (4) follows directly from the definition of a trace. From properties (2) and (3), the trace of α^(i) is a linear combination of a fixed set of coefficients of powers of α as shown in the following example:

EXAMPLE 3

From equation (7) where α^(i) 's defined by F_(B) (x)=x⁴ +x³ +1 appear in Table III and in this example ##EQU9## Since (as previously shown)

    Tr(α.sup.3)=Tr(α.sup.2)=Tr(α)=1

    and Tr(α.sup.0)=Tr(1)=0

    Tr(α.sup.i)=b.sub.3 +b.sub.2 +b.sub.1

The trace of each of the 16 elements are tabulated as follows:

    ______________________________________                                         i of a.sup.i                                                                            b.sup.3   b.sup.2                                                                              b.sup.i b.sup.O                                                                            Tr(a.sup.i)                               ______________________________________                                         *        0         0     0       0   0                                         0        0         0     0       1   0                                         1        0         0     1       0   1                                         2        0         1     0       0   1                                         3        1         0     0       0   1                                         4        1         0     0       1   1                                         5        1         0     1       1   0                                         6        1         1     1       1   1                                         7        0         1     1       1   0                                         8        1         1     1       0   1                                         9        0         1     0       1   1                                         10       1         0     1       0   0                                         11       1         1     0       1   0                                         12       0         0     1       1   1                                         13       0         1     1       0   0                                         14       1         1     0       0   0                                         ______________________________________                                    

The elements b₃ b₂ b₁ b₀ (representing α^(i) 's in equation (7)) of GF(2⁴) in Example 3 form a 4-dimensional vector space. Any set of linearly independent vectors which spans the vector space is a basis. The natural basis (previously discussed in connection with Table III.) is comprised of unit vectors α³ [1 0 0 0], α² [0 1 0 0], α[0 0 1 0] and 1 [0 0 0 1].

Berlekamp's Dual Basis

Berlekamp, motivated to significantly reduce the hardware complexity of spaceborne RS encoders [Perlman, et al., supra, and E. R. Berlekamp, "Bit-Serial Reed-Solomon Encoders," IEEE Transactions on Information Theory, Vol. IT 28, No. 6, pp. 869-874, November 1982] introduced parameter λ which results in another representation of RS symbols. The parameter λ is a field element where

    1, λ, λ.sup.2, . . . , λ.sup.r-1

is a basis in GF(2^(r)). Any field element that is not a member of subfield will form such a basis. If λ=α, a generator of the field, a natural basis results. Berlekamp's choice of λ and other independent parameters, such as the primitive polynomial known as the field generator polynomial, governing the representation of 8-bit RS symbols of a (255, 223) RS code was based solely on encoder hardware considerations detailed in Perlman, et al., supra. For a given basis

    {1, λ, λ.sup.2, . . . , λ.sup.r-1 }={λ.sup.k }

in GF(2^(r)), its dual basis {l_(j) }, also called a complementary or a trace-orthogonal basis, is determined. Each RS symbol corresponds to a unique representation in the dual basis

    {l.sub.0,l.sub.1, . . . ,l.sub.r-1 }={l.sub.j }

EXAMPLE 4

Consider the 16 field elements in Example 3. Members of subfields GF(2) and GF(2²) are

    α*, α.sup.0 and α*, α.sup.0, α.sup.5,α.sup.10

respectively. Let λ=α⁶ (one of 12 elements that is not a member of a subfield). Then

    {1, λ, λ.sup.2, α.sup.3 }={1, α.sup.6, α.sup.12, α.sup.9 }

    is a basis in GF(2.sup.4). Each field element has the following correspondence

    α.sup.i ⃡v.sub.0 l.sub.0 +v.sub.1 l.sub.1 +v.sub.2 l.sub.2 +v.sub.3 l.sub.3

where

    v.sub.k =Tr(λ.sup.k α.sup.i)=Tr(α.sup.6k+i)

The entries in column v₀ (coefficients of l₀) of Table IV below are

    v.sub.0 =Tr(α.sup.i)

as determined in Example 3. Whereas entries in column v₁ (coefficients of l₁) are

    v.sub.1 =Tr(α.sup.6+i)

which is column v₀ (excluding Tr(α*)) cyclically shifted upwards six places. The remaining columns are similarly formed.

                  TABLE IV                                                         ______________________________________                                         Elements in GF(2.sup.4) Represented in Basis                                   {l.sub.0, l.sub.1, l.sub.2, l.sub.3 }                                          i of a.sup.i                                                                          b.sub.3                                                                              b.sub.2                                                                              b.sub.1                                                                            b.sub.0                                                                             Tr(α.sup.i)                                                                     v.sub.0                                                                             v.sub.1                                                                            v.sub.2                                                                            v.sub.3                        ______________________________________                                         *      0     0     0   0    0      0    0   0   0                              0      0     0     0   1    0      0    1   1   1                              1      0     0     1   0    1      1    0   0   1                              2      0     1     0   0    1      1    1   0   0                              3      1     0     0   0    1      1    1   0   0                              4      1     0     0   1    1      1    0   1   0                              5      1     0     1   1    0      0    0   1   1                              6      1     1     1   1    1      1    1   1   1                              7      0     1     1   1    0      0    0   1   0   l.sub.2                    8      1     1     1   0    1      1    0   0   0   l.sub.0                    9      0     1     0   1    1      1    0   1   1                              10     1     0     1   0    0       0   1   0   0   l.sub.1                    11     1     1     0   1    0      0    1   1   0                              12     0     0     1   1    1      1    1   1   0                              13     0     1     1   0    0      0    1   0   1                              14     1     1     0   0    0      0    0   0   1   l.sub.3                    ______________________________________                                    

The dual basis of

    {λ.sup.k }={1,λ, λ.sup.2, λ.sup.3 } where λ=α.sup.6                                    ( 8)

as determined in Table IV is

    {l.sub.j }={l.sub.0, l.sub.1, l.sub.2,l.sub.3 }={α.sup.8, α.sup.10, α.sup.7, α.sup.14 }           (9)

The elements λ^(k) l_(j) ε GF(2⁴) and Tr(λ^(k) l_(j)) ε GF(2) appear, respectively, in the following "multiplication" tables where λ^(k) and l_(j) are arguments. ##EQU10## For the basis λ^(k) in (8) and its dual {l_(j) } in Equation (9) ##EQU11## as shown in the foregoing multiplication tables. Given any element α^(i) in GF(2⁴). Its components in {l_(j) } are readily computed as follows: ##EQU12## from property (3) of a trace and Equation (10).

Every basis in GF(2^(r)) has a dual basis. Furthermore, a one-to-one correspondence ##EQU13## exists from which Equation (11) follows with the appropriate change in the range of indices in Equations (10) and (11). The trace is used in modeling bit-serial multiplication in hardware of a fixed element expressed as a power of α (an RS symbol representing a coefficient of a generator polynomial g(x), subsequently discussed) and any RS symbol expressed in basis {l_(j) }. The resulting product is represented in basis {l_(j) }.

It is important to note that the one-to-one mapping(s) in Table IV of Example 4

    v.sub.0 l.sub.0 +v.sub.1 l.sub.1 +v.sub.2 l.sub.2 +v.sub.3 l.sub.3 →α.sup.i and α.sup.i →v.sub.0 l.sub.0 +v.sub.1 l.sub.1 +v.sub.2 l.sub.2 +v.sub.3 l.sub.3

can be achieved by either table look-up or by the following respective linear transformations: ##EQU14## The one-to-one onto mapping is an isomorphism where the operation of "addition" is preserved. Note that "multiplication" in basis {l_(j) } has not been defined.

In the section above on Mathematical Characterization of RS Symbols, the two primitive polynomials over GF(2⁴)

    F.sub.B (x)=x.sup.4 +x.sup.3 +1 and F.sub.G (x)=x.sup.4 +x+1

were discussed in connection with establishing representations for RS symbols. In RS coding terminology these are referred to as field generator polynomials. The subscript B in F_(B) (x) denotes the field generator polynomial with RS-symbol representation associated with Berlekamp encoder architecture. The subscript G in F_(G) (x) denotes the field generator polynomial with RS-symbol representation taken from GF(2⁴) defined by F_(G) (x). The associated encoder architecture is of the pre-Berlekamp or "conventional" type.

The Generator Polynomial and Structure of a Reed-Solomon Code with Conventional Type Architecture

The 2^(J) -1 (J-bit) symbols of the (N,K) RS codeword in (1) are GF(2^(J)) coefficients of the codeword polynomial

    c(x)=c.sub.N-1 x.sup.N-1 +C.sub.N-2 x.sup.N-2 +. . . +C.sub.2E x.sup.2E +C.sub.2E-1 x.sup.2E-1 +. . . +C.sub.0                    ( 12)

RS codes are cyclic and are completely characterized by a generator polynomial ##EQU15## Recall the cyclic property of RS codes introduced in the first section on RS code parameters and properties where every cyclic permutation of the symbols of a codeword results in a codeword. Also codewords of cyclic codes (a subset of linear codes) are mathematically described as codeword polynomials where every codeword polynomial C(x) expressed in Equation (12)) over GF(2^(J)) contains g(x) in Equation (13) as a factor. The generator polynomial (of an E symbol error-correcting RS code) g(x) of degree 2E, equal to number of symbols serving as check symbols, contains 2E consecutive nonzero powers of a primitive element β in GF(2^(J)) as roots. There is no restriction on which primitive element among φ(2^(J) -1) is chosen. Furthermore, though it has been well known that the 2E consecutive powers of a primitive element β^(s) could be

    (β.sup.s), (β.sup.s).sup.2, . . . , (β.sup.s).sup.2E

where (s, 2^(J) -1)=1. The value s=1 has invariably been used in practical applications. Berlekamp was the first to exploit the use of s>1 and more importantly 2E roots with a range of powers of j from b to b+2E-1 in (β^(s))^(j) in simplifying encoder hardware.

RS codes are a class of maximum distance separable (MDS) codes where the minimum Hamming distance

    D.sub.min =N-K+1=2E+1

is the maximum possible for a linear code over any field. "Separable" (in MDS) and "systematic" are synonymous terms for codes whose information symbols occupy leading adjacent positions and are followed by check symbols.

The Hamming weight of an RS codeword is the number nonzero symbols. The Hamming weight distribution of MDS (hence RS) codes is completely deterministic. Error probabilities can be computed from the Hamming weight distribution.

Encoding is the process of computing 2E check symbols over distinct subsets of K information symbols such that the N=K+2E symbols are coefficients of C(x) in (12) and g(x) is a factor of C(x).

Given the information polynomial

    I(x)=C.sub.n-1 x.sup.K-1 +C.sub.n-2 x.sup.k-2 +. . . +C.sub.2E ( 14)

check symbols are computed as follows: ##EQU16## From Equation (15)

    x.sup.2E I(x).tbd.r(x)mod g(x) C(x)=x.sup.2E I(x)+r(x).tbd.0 mod g(x) (17)

The polynomials x^(2E) I(x) and r(x) in Equation (16) are nonoverlapping and when "added" (where -r(x).tbd.r(x)) yield C(x) in (12). The generator polynomial g(x) is a factor of every (N,K) RS codeword polynomial C(x) as shown in (17).

A functional logic diagram of a conventional (N,K) RS encoder is given in FIG. 1. Assume the register (a cascade of 2E J-bit storage elements 1 through n of a RAM) of the FSR is initially cleared. With switches SW A and SW B in the up position, J-bit information symbols representing coefficients of I(x) are serially entered. An RS symbol is one argument for each of the multipliers M₀, M₁, . . . M_(2E-1), and the other argument is G₀, G₁, . . . G_(2E-1), respectively, where G_(i) is the coefficient of x^(i) of the codes generator polynomial. Symbol C_(N-1) is entered first. After the entry of C_(2E), the last information symbol, the check symbols (which represent coefficients of r(x)), reside in the register where C_(i) is stored in the register stage labeled x^(i). At this time, switches SW A and SW B are placed in the down position. The output of each multiplier is a bit-serial input to the respective modulo-two adders A₁ through A_(2E-1). Adder A_(in) is in the circuit only during the information input mode for reducing the input symbol, and the output of stage, x^(2E-1), are bit serially added during the information mode. See Equations (14) through (17).

The check symbols starting with C_(2E-1) are then delivered to the channel (appended to the information symbols) while the register is cleared in preparation for the next sequence of K information symbols. Multiplication of I(x) by x^(2E) is achieved by inputting the information symbols into the feedback path of the FSR in FIG. 1. The FSR divides x^(2E) I(x) by g(x) and reduces the result modulo g(x). The fixed multipliers

    G.sub.2E-1, G.sub.2E-2, . . . , G.sub.1, G.sub.0 ε GF(2.sup.J) (18)

in the respective interstage feedback paths of the FSR are coefficients of g(x) in Equation (13) (excluding G_(2E) =1). Note that

    g(x)=x.sup.2E +G.sub.2E-1 x.sup.2E-1 +G.sub.2E-2 x.sup.2E-2 +. . . +G.sub.1 x+G.sub.0

(and each scalar multiple of g(x)) is an RS codeword polynomial of lowest degree. A K information (J-bit) symbol sequence of K-1 0's (00 . . . 0) followed by the nonzero information symbol 1 (00 . . . 01) when encoded yields

    C(x)=Ox.sup.N-1 +Ox.sup.N-2 +. . . +Ox.sup.2E+1 +x.sup.2E +G.sub.2E-1 x.sup.2E-1 +. . . +G.sub.1 x+G.sub.0 =x.sup.2E +G.sub.2E-1 x.sup.2E-1 +. . . +G.sub.1 x+G.sub.0

since

x^(2E) .tbd.G_(2E-1) x^(2E-1) +. . . +G₁ x+G₀ mod g(x)

For each distinct multiplier in Equation (18), a ROM is addressed by the binary representation of a symbol appearing on the feedback path. Stored at the addressed location is the log (in binary) of the product of the symbol and the fixed multiplier reduced modulo 2^(J) -1. An antilog table stored in a ROM (which may be part of the same ROM storing logs) is subsequently accessed to deliver the binary form of the product.

EXAMPLE 5

Given J=4 and E=3, the parameters of a (15,9) RS code employing "conventional" encoder architecture shown in FIG. 1. The log.sub.β and binary representations of the RS symbols (i.e., field elements in GF(2⁴)) defined by the field generator polynomial

    F.sub.G (x)=x.sup.4 +x+1

appear in the following table:

    ______________________________________                                         i of β.sup.i                                                                           c.sub.3                                                                              c.sub.2     c.sub.1                                                                            c.sub.0                                     ______________________________________                                         *            0     0           0   0                                           0            0     0           0   1                                           1            0     0           1   0                                           2            0     1           0   0                                           3            1     0           0   0                                           4            0     0           1   1                                           5            0     1           1   0                                           6            1     1           0   0                                           7            1     0           1   1                                           8            0     1           0   1                                           9            1     0           1   0                                           10           0     1           1   1                                           11           1     1           1   0                                           12           1     1           1   1                                           13           1     1           0   1                                           14           1     0           0   1                                           ______________________________________                                    

Let the degree 6 (i.e., 2E) generator polynomial be ##EQU17## From (19), the fixed multipliers in the respective feedback paths of FIG. 1 (where G_(2E-1) =G₅) are

    {G.sub.5, G.sub.4, G.sub.3, G.sub.2, G.sub.1, G.sub.0 }={β.sup.10, β.sup.14, β.sup.4, β.sup.6, β.sup.9, β.sup.6 }

Since there are five distinct multipliers, five sets of ROM's are needed for "multiplication".

In Example 5, the field generator polynomial F_(G) (x) and the code generator polynomial g₁ (x) in Equation (19) completely characterize a (15,9) RS code which is 3 symbol error-correcting where D_(min) =2E+1=7. The 16 4-bit symbols appear in the table in Example 5 with their corresponding log.sub.β representations (i.e., i of β^(i)).

EXAMPLE 6

Consider the following (15,9) RS codewords with symbols expressed in log.sub.β form (i.e., i of β^(i)) as given in the table in Example 5. The code's generator polynomial is g₁ (x) in Equation (19).

    __________________________________________________________________________     Codeword                                                                             C.sub.14                                                                          C.sub.13                                                                          C.sub.12                                                                          C.sub.11                                                                          C.sub.10                                                                          C.sub.9                                                                          C.sub.8                                                                          C.sub.7                                                                          C.sub.6                                                                          C.sub.5                                                                          C.sub.4                                                                          C.sub.3                                                                          C.sub.2                                                                          C.sub.1                                                                          C.sub.0                                 C.sup.0                                                                              *  *  *  *  *  * * * * * * * * * *                                       C.sup.1                                                                              *  *  *  *  *  * * *  0                                                                               10                                                                               14                                                                                4                                                                                6                                                                                9                                                                                6                                      C.sup.2                                                                              *  *  *  *  *  *  0                                                                               10                                                                               14                                                                                4                                                                                6                                                                                9                                                                                6                                                                               * *                                       C.sup.3                                                                              *  *  *  *  *  *  0                                                                               10                                                                                3                                                                                2                                                                                8                                                                               14                                                                               *  9                                                                                6                                      C.sup.4                                                                              10 10 10 10 10 10                                                                               10                                                                               10                                                                               10                                                                               10                                                                               10                                                                               10                                                                               10                                                                               10                                                                               10                                      __________________________________________________________________________

C¹ (x)=g₁ (x), a codeword polynomial of least degree. C⁰ (x) is the codeword polynomial whose coefficient string C^(O) is * * . . . * (where β*=0000). Since RS codes are linear or group codes, C⁰ (of an appropriate length) is a codeword of every (N,K) RS code over GF(p^(r)). Furthermore, every sequence of N identical symbols such as C⁴ is an RS codeword. Codeword C² is a cyclic permutation (two places to the left) of C¹. Thus

    c.sup.2 (x)=x.sup.2 c.sup.1 (x)=x.sup.2 g.sub.1 (x)

(where powers of x are necessarily reduced modulo 15). Codeword C³ is a linear combination of C¹ and C² and

    c.sup.3 (x)=(x.sup.2 +1)g.sub.1 (x).

Clearly

    g.sub.1 (x)|C.sup.0 (x)

and dividing C⁴ (x) by g₁ (x) leads to the Euclidean form

    c.sup.4 (x)=(β.sup.10 x.sup.8 +x.sup.7 +β.sup.9 x.sup.6 +β.sup.2 x.sup.5 +β.sup.6 x.sup.4 +β.sup.8 x.sup.3 +β.sup.6 x.sup.2 +β.sup.3 x+β.sup.4)g.sub.1 (x)

The Hamming distance between distinct pairs of codewords are listed in the following table.

    ______________________________________                                         Codewords    Hamming Distance                                                  ______________________________________                                         C.sup.0, C.sup.1                                                                             7                                                                C.sup.0, C.sup.2                                                                             7                                                                C.sup.0, C.sup.3                                                                             8                                                                C.sup.0, C.sup.3                                                                            15                                                                C.sup.1, C.sup.2                                                                             8                                                                C.sup.1, C.sup.3                                                                             7                                                                C.sup.1, C.sup.4                                                                            14                                                                C.sup.2, C.sup.3                                                                             7                                                                C.sup.2, C.sup.4                                                                            14                                                                C.sup.3, C.sup.4                                                                            14                                                                ______________________________________                                    

Refer to FIG. 1 and the (15,9) RS code discussed in Examples 5 and 6. The six register stages x⁵, x⁴, x³, x², x, 1 are initially cleared. Successive states of the six stages when encoding the following sequence of information symbols

    ______________________________________                                         {C.sub.14, C.sub.13, . . . , C.sub.9, C.sub.8, C.sub.7, C.sub.6 } = {*,        *, . . . , *, 0, 10, 3} are                                                    j of c.sub.j                                                                           Symbol     x.sup.5                                                                              x.sup.4                                                                              x.sup.3                                                                            x.sup.2                                                                              x   1                                 ______________________________________                                         14      *          *     *     *   *     *   *                                 13      *          *     *     *   *     *   *                                 .       .          .     .     .   .     .   .                                 .       .          .     .     .   .     .   .                                 .       .          .     .     .   .     .   .                                 9       *          *     *     *   *     *   *                                 8       0          10    14     4  6     9   6                                 7       10         14     4     6  9     6   *                                 6       3           2     8    14  *     9   6                                                    ↑                                                                        C.sub.5                                                     ______________________________________                                    

The states of the register stages in a given row reflects the entry of the corresponding information symbol. When entering an information symbol, it is "added" by Adder A^(in) to the content of stage x⁵ (i.e., x^(2E-1)) as it is shifting out (to the left in FIG. 1) and the resulting symbol is simultaneously "multiplied" by each feedback coefficient. This string of symbols is (vector) "added" to the (left) shifted contents of the register. It is this result that corresponds to the states of the register after entering an information symbol. The following events occur when entering the last information symbol C₆ in deriving C³ given on page 28. ##STR1## As shown in FIG. 1 "addition" of symbols is done bit-serially during shifting. Whereas each of the five different "multiplications" requires a log and antilog table stored in a ROM as previously discussed. The remainder upon dividing x⁶ (x⁸ +β¹⁰ x⁷ +β³ x⁶) by g₁ (x) in (19) verifies the (hardware) encoding result. ##STR2## The quotient

    0*0 ⃡x.sup.2 +1 and g.sub.1 (x)

are factors of C³ (x).

The Generator Polynomial and Structure of a Reed-Solomon Code with Berlekamp Architecture

The generator polynomial g(x) used in Berlekamp (N,K) RS encoder architecture is a self-reciprocal polynomial of degree 2E over GF(2^(J)). As in the design of conventional (N,K) RS encoder architecture GF(2^(J)) is defined by a field generator polynomial of degree J over GF(2). The generator polynomial ##EQU18## where α and γ=α^(s) are primitive elements in GF(2^(J)). (A positive integer value of s is chosen whereby (s, 2^(J) -1)=1)). In contrast with g(x) in Equation (13) associated with conventional architecture γ is any primitive element and j ranges from b to b+2E-1. The number of multiplications per symbol shift is approximately halved by selecting a g(x) with E pairs of reciprocal roots.

Reciprocal root pairs in Equation (20) are

    γ.sup.b+i γ.sup.(b+2E-1)-i =1=γ.sup.N for 0≦i<E (21)

and

    2b+2E-1=N=2.sup.J -1b=2.sup.J-1 -E                         (22)

from which b and b+2E-1 are determined given J (bits per symbol) and E (symbol error-correction capability)

EXAMPLE 7

Given J=4 and E=3, parameters of a (15,9) RS code employing Berlekamp encoder architecture shown in FIG. 2. The log.sub.α and binary representations of the RS symbols in GF(2⁴) defined by the field generator polynomial

    F.sub.B (x)=x.sup.4 +x.sup.3 +1

are given in TABLE I. A degree 6 generator polynomial is derived for γ=α⁴ in Equation (20). From Equations (21) and (22) ##EQU19## Note that the coefficient string of g₂ (x) in (23) is palindromic since

    x.sup.6 g.sub.2 (1/x)=g.sub.2 (x)

is a self-reciprocal polynomial over GF(2⁴). Thus ##EQU20## represent the fixed multipliers. A multiplier G_(i) =α⁰ =1 corresponds to a wire (in either conventional or Berlekamp encoder hardware) and does not incur a cost. In Berlekamp architecture G_(O) =(G_(2E)) necessarily equals α⁰ =1 and there are at most E distinct nonzero fixed multipliers excluding α⁰. Whereas in conventional architecture there are at most 2E distinct nonzero fixed multipliers.

The number of distinct possible self-reciprocal generator polynomials g(x) of degree 2E is φ(2^(J) -1)/2 for any J and E where (2^(J) -1)-2E≧3. For each γ(α^(s) in Equation (20)) its reciprocal γ⁻¹ (i.e., α^(-s) =α^(N-s)) yields the same g(x). In this example there are φ(15)=8 distinct choices in selecting γ resulting in four distinct possible self-reciprocal generator polynomials g₂ (x).

As is the case for conventional (encoder) architecture, coefficients of the generator polynomial associated with Berlekamp architecture are represented as

    α.sup.i =b.sub.r-1 α.sup.r-1 +b.sub.r-2 α.sup.r-2 +. . . +b.sub.0 where b.sub.j ε GF (2)

in GF(2^(r)). See Table I and Example 7 for r=4. However, unlike the case for conventional architecture, information symbols and check symbols are represented in basis {l_(j) } the dual basis of {λ^(k) } discussed above in the section on Berlekamp's Representation of RS Symbols Using the Concept of Trace and Berlekamp's Dual Basis. ##EQU21## for a selected element λ in GF(2^(r)) where λ is not a member of a subfield.

Referring to FIG. 2, a linear binary matrix 10 (i.e., an array of Exclusive-OR gates) with J inputs and E+2 outputs is used to perform the multiplication of a J-bit RS symbol (in the feedback path) stored in a z register 12 with each of the coefficients

    G.sub.0 =(G.sub.2E), G.sub.1 =G.sub.2E-1, . . . , G.sub.E-1 =G.sub.E+1, G.sub.E

of g₂ (x). The components of z (an RS symbol represented in {l_(j) }) are

    z.sub.0, z.sub.1, . . . z.sub.j-1

Note that substituting z_(i) for v_(i) gives ##EQU22## where α^(i) is the corresponding representation of the RS symbol (i.e., field element) z. Thus, as in Equation (11), z_(k) in {l_(j) } is computed by ##EQU23## At a given time interval the representation of a field element z in basis {l_(j) } is entered and stored in register z. The respective outputs of the linear binary matrix are ##EQU24## These outputs represent

    z.sub.0.sup.(0), z.sub.0.sup.(1), . . . , z.sub.0.sup.(E)  ( 25)

the first component (bit) in basis {l_(j) } of each of the products

    zG.sub.0, zG.sub.1, , . . . , zG.sub.E                     ( 26)

respectively. Condensed symbolism associated with expressions of the form appearing in Equations (24), (25) and (26) proposed by Berlekamp appear in Perlman and Lee, supra. The condensed symbolism is introduced here in an attempt to complete the description of the operation of the linear binary matrix.

    {z.sub.0.sup.(l) } for 0≦l≦E

is the equivalent of Equation (25). Whereas {zG₁ } is a condensed representation of Equation (26).

    T.sub.l (z)=Tr(zG.sub.l)

denotes J successive sets of E simultaneous outputs (bit per output) in computing {zG_(l) } in basis {l_(j) }. Since zG_(l) is representable in basis {l_(j) } the dual basis of {λ^(k) } ##EQU25## At (time interval) k=0, the simultaneous outputs {T_(l) } in Equation (24) are {z₀ .sup.(l) } (in Equation (25)), the first component (bit) of each of the products {zG_(l) } in Equation (26). At k=1, the stored symbol z (in FIG. 2) is replaced by λz where λz is derived from z (as will be shown). The simultaneous outputs {T_(l) } are {z₁.sup.(l) }, the second component (bit) of each of the products {zG_(l) }. Similarly at k=2, λz is replaced by λ(λz) and {T_(l) } yields the third component of {G_(l) } and so on.

The form of the {T_(l) } functions follows from Equation (27) and is ##EQU26## For every z, the output T_(l) for a given l is the sum modulo 2 (Exclusive-OR) of those components z_(j) 's in the dual basis for which

    Tr(l.sub.j G.sub.l)=1

The output Tr(λ^(J) z) in FIG. 2 which is fed back to the z register is used in deriving λz. The field element z may be represented as α^(i) or in basis {l_(j) } in vector form as

    z=Tr(z), Tr(λz), . . . , Tr(λ.sup.J-1 z)

where

    Tr(λ.sup.k z)=Tr(λ.sup.k α.sup.i)=z.sub.k

and

    z=Tr(λz), Tr(λ.sup.2 z), . . . , Tr(λ.sup.J z)

By shifting the bits stored in the z register in FIG. 2 upward one position and feeding back the output (of the linear binary matrix) Tr(λ^(J) z), λz is derived from z. This corresponds to

    Z.sub.i ←z.sub.i+1 for 0≦i<J z.sub.J-1 ←z.sub.J =Tr(λ.sup.J z)

EXAMPLE 8

The design of the linear binary matrix for the (15,9) RS code introduced in Example 7 is the subject of this example. One argument for each {T_(l) } function is the set of distinct unit vectors in {l_(j) }

    {l.sub.0, l.sub.1, l.sub.2, l.sub.3 }={α.sup.8, α.sup.10, α.sup.7, α.sup.14 }

expressed as α^(i). See Table IV where components z_(i) are substituted for coefficients v_(i). The other argument is a distinct coefficient

    {G.sub.0, G.sub.1, G.sub.2, G.sub.3 }={α.sup.0, α.sup.4, α.sup.2, α.sup.11 }

of the self-reciprocal generator polynomial g₂ (x) in Equation (23).

The following table yields the desired {T_(l) } linear functions of the z_(i) components.

    ______________________________________                                         j        0      1      2    3                                                  l.sub.j G.sub.0                                                                         α.sup.8                                                                         α.sup.10                                                                        α.sup.7                                                                       α.sup.14                                     Tr(l.sub.j G.sub.0)                                                                     1      0      0    0     T.sub.0 = z.sub.0                            l.sub.j G.sub.1                                                                         α.sup.12                                                                        α.sup.14                                                                        α.sup.11                                                                      α.sup.3                                      Tr(l.sub.j G.sub.1)                                                                     1      0      0    1     T.sub.1 = z.sub.0 + z.sub.3                  l.sub.j G.sub.2                                                                         α.sup.10                                                                        α.sup.12                                                                        α.sup.9                                                                       α                                            Tr(l.sub.j G.sub.2)                                                                     0      1      1    1     T.sub.2 = z.sub.1 + z.sub.2 + z.sub.3        l.sub.j G.sub.3                                                                         α.sup.4                                                                         α.sup.6                                                                         α.sup.3                                                                       α.sup.10                                     Tr(l.sub.j G.sub.3)                                                                     1      1      1    0     T.sub.3 = z.sub.0 + z.sub.1                  ______________________________________                                                                           + z.sub.2                               

The output of the linear binary matrix Tr(λ^(J) α^(i)) (where J=4 and λ=α⁶), which is fed back when computing λz, is derived from Table IV as follows:

    ______________________________________                                         z.sub.0                                                                             z.sub.1                                                                               z.sub.2                                                                               z.sub.3                                                                             α.sup.i                                                                        λ.sup.4 α.sup.i                                                   = α.sup.9+i                                                                         Tr(λ.sup.4 α.sup.i)      ______________________________________                                         1    0      0      0    α.sup.8                                                                        α.sup.2                                                                             1                                     0    1      0      0    α.sup.10                                                                       α.sup.4                                                                             1                                     0    0      1      0    α.sup.7                                                                        α    1                                     0    0      0      1    α.sup.14                                                                       α.sup.8                                                                             1                                     ______________________________________                                         Thus, Tr(λ.sup.4 α.sup.i) = z.sub.0 + z.sub.1 + z.sub.2 +         z.sub.3 (= z.sub.4)                                                            ______________________________________                                    

This completes the logical description of the linear binary array for the (15,9) RS code presented in Example 7. An implementation is given in FIG. 3 in the form of a functional logic diagram. By (Exclusive-OR) gate sharing, the total of number of gates is reduced to 5.

Berlekamp architecture eliminates all firmware required by conventional architecture for multiplying a field element (RS Symbol) by a set of fixed field elements (corresponding to coefficients of the generator polynomial). The firmware is replaced by the linear binary array 10 whose complexity is governed by the selected field generator polynomial, the basis {λ^(k) } whose dual basis is {l_(j) }, and the self-reciprocal generator polynomial. For a given E (symbol error-correction capability) these selections are totally independent. Following is the total number of ways these parameters can be chosen for a J (bits per symbol) of 4 and 8.

    ______________________________________                                                               Combined                                                 No. of Ways of Selecting                                                                             Independent                                              J       F.sub.B (x)                                                                            g.sub.2 (x)                                                                               λ                                                                            Selections                                     ______________________________________                                         4        2       4          12  96                                             8       16      64         240  245, 760                                       ______________________________________                                    

In GF(2⁸), the subfield GF(2⁴) of 16 elements contains GF(2²) which contains GF(2). Thus λ can be selected from 240 elements in GF(2⁸) each of which is not a member of GF(2⁴). For a given F_(B) (x) and g₂ (x) Berlekamp used a computer search to obtain a complexity profile of the linear binary array for a (255,223) RS code. This enabled him to select a region where λ, F_(G) (x) and g₂ (x) led to a linear binary array of minimal complexity.

In FIG. 2, a y register 14 is essentially an extension of register section 2E-1 and serves as a staging register. After the products {zG_(l) } have been determined, register z is reloaded with the contents of y the next symbol to be fed back. Until all information symbols have been entered and simultaneously delivered to the channel, the y input is the bit-by-bit Exclusive-OR of the bits comprising the information symbol being entered and the bits of the symbol exiting register section S_(2E-j). After the last information symbol has been entered, the mode is changed by a mode switch MSW from information to check, and the 2E check symbols are delivered bit-serially to the channel. Register sections S₀, S₁, . . . , S_(2E-1) reside in RAM's. The linear binary array 10 allows simultaneous bit-serial multiplication and addition sum modulo 2 with corresponding components of bit shifted symbols in each register section. It effectively realizes bit-serial multiplication of an arbitrary RS symbol with a set of fixed multipliers. The register stages S₀, S₁, . . . S_(2E-1) play the same role as corresponding register stages in FIG. 1. Similarly, addersA₁, . . . , A_(2E-1) and A_(in) play a similar role as adders in FIG. 1.

EXAMPLE 9

    ______________________________________                                                  z   λz   λ.sup.2 z                                                                      λ.sup.3 z                               ______________________________________                                         z.sub.0    0     0           0    1                                            z.sub.1    0     0           1    1                                            z.sub.2    0     1           1    0                                            z.sub.3    1     1           0    0                                            ______________________________________                                    

Columns (left to right) are successive contents of the z register after first nonzero information symbol is entered.

    __________________________________________________________________________     C.sub.14                                                                         0   0 0 0 0   0 0 0 0 0   0 0 0 0 0 0   α*                             C.sub.13                                                                         0   0 0 0 0   0 0 0 0 0   0 0 0 0 0 0   α*                             .     .         .           .       .     .                                    .     .         .           .       .     .                                    .     .         .           .       .     .                                    C.sub.6                                                                          0   0 0 1 0   0 0 1 0 0   0 1 0 0 0 1   α.sup.14                       C.sub.5                                                                          1         1   1     1 1   0   1 1 0 1   α.sup.3                        C.sub.4                                                                          1         1   0     1 0   0   1 0 0 1   α                              C.sub.3                                                                          0         0   1     0 1   0   0 1 0 0   α.sup.10                       C.sub.2                                                                          1         1   0     1 0   0   1 0 0 1   α                              C.sub.1                                                                          1         1   1     1 1   0   1 1 0 1   α.sup.3                        C.sub.0                                                                          0         0   0     0 0   0   0 0 0 1   α.sup.14                         ↑   ↑     ↑       ↑                                    T.sub.0 (z)                                                                              T.sub.1 (z) T.sub.2 (z)   T.sub.3 (z)                              __________________________________________________________________________

The foregoing tables are associated with the encoding of the information symbol sequence

    ______________________________________                                         C.sub.14                                                                            C.sub.13                                                                              . . .  C.sub.6                                                     0    0      . . .  1   Info. Symb. in Basis {l.sub.j } Expressed in            ______________________________________                                                                Hex.                                               

The encoder in FIG. 2 (where J=4 and E=3) is initially cleared and in the information mode. The last symbol C₆ to enter the encoder is the only nonzero symbol among nine information symbols. Previous all zero (0 0 0 0) symbols have no affect on the initial zero states of the registers y and z and the register sections S₅, S₄, . . . , S₀. Symbol C₆ is entered and stored (via the y register) in the z register. (Since register section S₅ contains 0 0 0 0 upon entering C₆, the bit-by-bit Exclusive-OR of the two symbols leaves C₆ unchanged). The contents of the z register after C₆ is entered is shown in the first of the foregoing tables under column heading z, where

    z.sub.0 =z.sub.1 =z.sub.2 =0 and z.sub.3 =1

The following {T_(l) } functions derived in Example 8 ##EQU27## are the first set of outputs (with the current components of the z register as arguments). These are ##EQU28## the first respective components of check symbols C₀, C₁, . . . , C₅ in the second table of this example. As previously discussed the {T_(l) } outputs in (29) occur simultaneously. The last (bit) entry in the column identified as

    T.sub.0 (z)=Tr(zG.sub.0)

is the first component of check symbol C_(O). After computing the first components of T_(l) (z)=Tr(zG_(l)) in (28) for j=0, the contents of the z register is shifted upward one (bit) position with z₃ replaced by ##EQU29## The new contents of the z register appears in the first table of this example under the column heading λz. The second components of T_(l) (z)=Tr(zG_(l)) in (28) for j=1 are similarly computed and tabulated in the lower table. The second component of check symbol C_(O) when first computed (along with the second components of all the other check symbols) is the last bit in the column identified as

    T.sub.1 (z)=Tr(λzG.sub.0)

Encoding continues until all four components of each check symbol have been computed in basis {l_(j) } as shown in the lower table. The information symbols and check symbols of the (15,9) RS codeword are in basis {l_(j) }. Their corresponding α^(i) representation appears in the rightmost column of the second table in accordance with TABLE IV. Consider the following representations of the resulting codeword.

    __________________________________________________________________________     C.sub.15                                                                          C.sub.14                                                                          . . .                                                                             C.sub.7                                                                           C.sub.6                                                                           C.sub.5                                                                           C.sub.4                                                                           C.sub.3                                                                           C.sub.2                                                                           C.sub.1                                                                           C.sub.0                                                                           Representation                                0  0  . . .                                                                             0  1  D  9  4  9  D  1  Basis {l.sub.j } in Hex                       α*                                                                          α*                                                                          . . .                                                                             α*                                                                          α.sup.14                                                                    α.sup.3                                                                     α                                                                           α.sup.10                                                                    α                                                                           α.sup.3                                                                     α.sup.14                                                                    {α.sup.i }                              __________________________________________________________________________

The representation in powers of α corresponds to the polynomial

    P(x)=α.sup.14 x.sup.6 +α.sup.3 x.sup.5 +αx.sup.4 +α.sup.10 x.sup.3 +αx.sup.2 +α.sup.3 x+α.sup.14

Comparing P(x) with the code's generator polynomial

    g.sub.2 (x)=x.sup.6 +α.sup.4 x.sup.5 +α.sup.2 x.sup.4 +α.sup.11 x.sup.3 +α.sup.2 x.sup.2 +α.sup.4 x+1

reveals that P(x) is a scalar multiple of g₂ (x)--i.e.,

    P(x)=α.sup.14 [g.sub.2 (x)]=C(x)

a codeword polynomial.

STATEMENT OF THE INVENTION

Mappings between codewords of two distinct (N,K) Reed-Solomon codes over GF(2^(J)) having selected two independent parameters: J, specifying the number of bits per symbol; and E₁, the symbol error correction capability of the code, is provided. The independent parameters J and E yield the following: N=2^(J) -1, total number of symbols per codeword; 2E, the number of symbols assigned a role of check symbols; and K=N-2E, the number of code symbols representing information, all within a codeword of an (N,K) RS code over GF(2^(J)). Having selected those parameters for encoding, the implementation of a decoder is governed by: 2^(J) field elements defined by a degree J primitive polynomial over GF(2) denoted by F(x); a code generator polynomial of degree 2E containing 2E consecutive roots of a primitive element defined by F(x); and, in a Berlekamp RS code, the basis in which the RS information and check symbols are represented.

The process for transforming words R_(B) in a conventional code for decoding the words R_(G) having possible errors to codewords C_(G) free of error is accomplished in a sequence of transformations leading to decoding R_(G) to C_(G), and inverse transformations of those same transformations in a reverse sequence leading to converting the codewords C_(G) to codewords C_(B), thus decoding the Berlekamp codewords R_(G). Thus, the corrected codewords C_(G) are reverse transformed symbol-by-symbol back into codewords C_(B) in the Berlekamp code. The uncorrected words in the Berlekamp code and the conventional code are here denoted by the letter R with subscripts B and G, respectively, because they are with possible error. Until corrected, they are not denoted codewords by the letter C with first the subscript G and later in the process the subscript B. The codewords C_(G) are not information, because in the process of transformation from R_(B) to R_(G) for decoding (i.e., correcting errors) they have undergone permutation. To retrieve the information, the codewords C_(G) must be transformed back into codewords C_(B) by inverse transformation in a reverse order of process steps.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a conventional (N,K) RS encoder.

FIG. 2 illustrates an (N,K) RS endecoder utilizing Berlekamp's architecture.

FIG. 3 is a logic diagram of a linear binary matrix for a (15,9) RS Berlekamp encoder.

FIGS. 4a through 4d illustrate a flow chart of a sequence of transformations and their respective inverses in reverse order, in accordance with the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The Mathematical Equivalence of (N,K) Reed-Solomon Codes

Before proceeding with a detailed description of the invention, the transformational equivalence of Berlekamp and conventional RS codes for (N,K)=(15,9) will be described as one example of an (N,K) RS code GF(w⁴). Then a succession of transformations will be described that, in accordance with the present invention, map any given codeword of a (255,223) RS Berlekamp code B to a codeword of a (255,223) RS code. (Conventional code G). This allows the use of one ground decoder (associated with code G) which is conservatively an order of magnitude more complex than its associated encoder to serve as a decoder for the two distinct (255,223) RS codes. The succession of transformations that map a received word (which may be erroneous) originating from code B to one of code G (with the number of erroneous symbols unchanged) is completed before decoding. After decoding where symbol errors, if any, have been corrected, the entire codeword (now a member of code G) is subjected to the inverse of each of the above transformations in reverse order to recover the received codeword in Code B in order to recover the codeword originating from code B (corrected if originally erroneous). A received word originating from code G is decoded directly. B in code B denotes Berlekamp's representations associated with Berlekamp encoder architecture. Whereas G in code G denotes representations associated with pre-Berlekamp or "conventional" architecture.

The mathematical description of two (15,9) RS codes are summarized as follows: ##EQU30## Each (of the 9) 4-bit information symbols of any (15,9) RS code is independently selected among 16 possible symbols in GF(2⁴) defined by the code's field generator polynomial. Each (of the 6) check symbol is a linear combination of a distinct subset of information symbols and thus are defined by the information symbols and, therefore, dependent. Thus the size of a (15,9) RS codeword dictionary is

    (2.sup.4).sup.9 =2.sup.36 ≈6.87×10.sup.10 codewords

There is a one-to-one correspondence between the codewords of two different (N,K) RS codes.

Given the information symbol sequence represented in hexadecimal

    ______________________________________                                         i of C.sub.i                                                                           14    13     12   11   10   9    8    7    6                           {l.sub.j } in Hex                                                                       7     0      0    0    0   0    0    0    0                           ______________________________________                                    

encoding the information symbol sequence using an encoder with Berlekamp architecture results in the Code B codeword

    __________________________________________________________________________     i of C.sub.i                                                                          14 13                                                                               12                                                                               11                                                                               10                                                                               9 8 7 6 5 4 3 2 1 0                                          {l.sub.j } in Hex                                                                      7  0                                                                                0                                                                                0                                                                                0                                                                               0 0 0 0 7 A C 6 C A                                          __________________________________________________________________________

where each 4-bit symbol is a basis {l_(j) } symbol expressed in Hexadecimal.

The first transformation to be applied to a Code B codeword is the mapping of each symbol from its basis {l_(j) } representation to its corresponding

    α.sup.i =b.sub.3 α.sup.3 +b.sub.2 α.sup.2 +b.sub.1 α.sup.1 +b.sub.0 α.sup.0

representation as given in TABLE IV. The log.sub.α of α^(i) (i.e., i) is (and has been) used to represent the binary symbol b₃ b₂ b₁ b₀ where appropriate.

EXAMPLE 10

Consider the Code B codeword

    __________________________________________________________________________     k of Ck                                                                               14                                                                               13                                                                               12                                                                               11                                                                               10                                                                               9 8 7 6 5 4 3 2 1 0                                           {l.sub.j } in Hex                                                                      7                                                                                0                                                                                0                                                                                0                                                                                0                                                                               0 0 0 0 7 A C 6 C A                                           __________________________________________________________________________

By table look-up or the linear transformation discussed above under the heading "Berlekamp's Representation of RS Symbols Using the Concept of a Trace," the basis {l_(j) } symbols are transformed to their respective α^(i) representations to yield the Code B.1 codeword

    __________________________________________________________________________     k of Ck                                                                               14                                                                               13                                                                               12                                                                               11                                                                               10                                                                               9 8 7 6 5 4 3  2                                                                               1 0                                           {l.sub.j } in Hex                                                                      7                                                                                0                                                                                0                                                                                0                                                                                0                                                                               0 0 0 0 7 A C  6                                                                               C A                                           i of α.sup.i                                                                     0                                                                               * * * * * * * * 0 4 2 11                                                                               2 4                                           __________________________________________________________________________

Note that Code B.1 is a (15,9) RS code with the following mathematical description: ##EQU31## Code B.1 has the same field and code generator polynomial (which is self-reciprocal) as Code B. The encoder associated with Code B.1 is of the "conventional" type, a codeword denoted C_(B).1 (x) in the following equation.

    C.sub.B.1 (x)=x.sup.14 +x.sup.5 +α.sup.4 x.sup.4 +α.sup.2 x.sup.3 +α.sup.11 x.sup.2 +α.sup.2 x+α.sup.4 (30)

is from Example 10 a codeword polynomial belonging to Code B.1 which contains g₂.1 (x)=g₂ (x) as a factor. The symbol-by-symbol transformation from the basis {l_(j) } representation in binary to its corresponding α^(i) representation in binary (i.e., αbasis} is ##EQU32## is the linear transformation matrix. For example

    [1 1 1 0]T.sub.lα =[0 0 1 1]=α.sup.12

as may be verified in TABLE IV.

The second transformation in the sequence is related to the translation of the powers of the roots in g₂.1 (x) (=g₂ (x)) which was first derived in R. L. Miller and L. J. Deutsch, "Conceptual Design for a Universal Reed-Solomon Decder," IEEE Transactions on Communications," Vol. Com-29, No. 11, pp. 1721-1722, November 1981, for the general case. Given ##EQU33## where γ=α⁴. Changing the argument from x to γ⁴ x yields ##EQU34## where (γ⁴)⁶ =γ²⁴.tbd.9 mod 15 and γ⁴ =α.

The generator polynomial g₂.2 (x) derived from g₂.1 (x) defines the (15,9) RS Code B.2.

Given C_(B).1 (x) a Code B.1 codeword polynomial. Thus

    g.sub.2.1 (x)|C.sub.B.1 (x) and C.sub.B.1 (x)=u(x)g.sub.2.1 (x) over GF(2.sup.4)

defined by F_(B).1 (x) (=F_(B) (x)). Then

    γ.sup.-9 C.sub.B.1 (αx)=u(αx)[γ.sup.-9 g.sub.2.1 (αx)]=u(αx)g.sub.2.2 (x)

and

    C.sub.B.1 (αx)=γ.sup.9 u(αx)g.sub.2.2 (x)

since g₂.2 (x)|C_(B).1 (αx), C_(B).1 (αx) is a Code B.2 codeword polynomial.

Code B.2 is a (15,9) RS code with the following mathematical description: ##EQU35## The encoder associated with Code B.2 is of the "conventional" type. The generator polynomial g₂.2 (x) provides a test for any codeword polynomial C_(B).2 (x) derived from the transformation C_(B).1 (αx). Dividing C_(B).2 (x) by g₂.2 (x) in (31) over GF(2⁴) defined by F_(B).2(x) =(F_(B) (x)) results in a zero remainder.

EXAMPLE 11

From Example 10 and Equation (30) ##EQU36## It may be verified that

    C.sub.B.2 (x)=(α.sup.14 x.sup.8 +α.sup.2 x.sup.7 +α.sup.7 x.sup.6 +α.sup.4 x.sup.5 +α.sup.4 x.sup.4 +α.sup.3 x.sup.3 +αx.sup.2 +α.sup.2 x+α.sup.10)g.sub.2.2 (x)

over GF(2⁴) defined by F_(B).2 (x) (=F_(B) (x)). The generator polynomial g₂.2 (x) appears in (31).

The outcomes of successive transformations, thus far, on codeword (symbols) C_(B) in Example 10 are:

    __________________________________________________________________________     k of Ck                                                                               14                                                                               13                                                                               12                                                                               11                                                                               10                                                                               9 8 7 6 5 4 3  2                                                                               1 0                                           __________________________________________________________________________     C.sub.B                                                                                7                                                                                0                                                                                0                                                                                0                                                                                0                                                                               0 0 0 0 7 A C  6                                                                               C A                                           C.sub.B.1                                                                              0                                                                               * * * * * * * * 0 4 2 11                                                                               2 4                                           C.sub.B.2                                                                             14                                                                               * * * * * * * * 5 8 5 13                                                                               3 4                                           __________________________________________________________________________

The binary symbols of C_(B) are in basis {l_(j) } represented in hexadecimal. Whereas the binary symbols in C_(B).1 and C_(B).2 are represented by i of α^(i) in the same GF(2⁴) where F_(B) (α)=0.

The transformation related to the translation of the powers of the roots involves a change in the magnitude of s_(k) in α^(SK) of a coefficient of x^(k) (when α^(SK) and k are nonzero) which is dependent upon the magnitude of k. An all zeros coefficient (α*) and C_(O) the constant term of a codeword polynomial are unaffected. Given the fixed vector

    14, 13, . . . , 1, 0

representing the powers of x of any (15,9) codeword polynomial and

    s.sub.14, s.sub.13, . . . , s.sub.1, s.sub.0

the log.sub.α of respective coefficients of codeword polynomial C_(B).1 (x). Then the log.sub.α of the coefficient of x^(k) of C_(B).2 (x) (i.e., C_(B).1 (αx)) is

    k+s.sub.k mod 15 and for s.sub.k =*, k+*=*

The description of the succession of transformations for the one-to-one mapping of C_(B) (x) to C_(G) (x) continues. A change in the primitive element α is required in ##EQU37## in preparation for the one-to-one mapping of RS symbols (i.e., field elements) in GF(2⁴) defined by F_(B) (x) in (5) to RS symbols in GF(2⁴) defined by F_(G) (x) in (6). See Table III.

An element α^(y) is sought whereby

    (α.sup.y).sup.4 =α.sup.13 or 4y.tbd.13 mod 15  (32)

From number theory, for integers b, c and n

    by.tbd.c mod n

has solution(s) in y if and only if (b,n)|c and the number of distinct solutions is (b,n). Since (4,15)=1 there is a unique solution for y in (32), namely, y=7. Clearly (7,15)=1 and α⁷ is primitive.

Recall that α is a primitive root of F_(B) (x) and that α⁴ among the candidates of primitive elements

    α(or α.sup.14), α.sup.2 (or α.sup.13), α.sup.4 (or α.sup.11), α.sup.7 (or α.sup.8)

was selected in deriving one of 4 distinct self-reciprocal generator polynomials, namely, g₂ (x) in (23). The choice is one factor affecting the complexity of the linear binary array in FIG. 2. Since i of the α^(i) candidates is necessarily relatively prime to 15 (i.e., (i,n)=1), a unique solution exists for the congruence in (32) where each i is substituted for 4. The foregoing arguments are applicable in formulating a transformation between any two distinct (N,K) RS codes involving a change in the primitive element. N-K consecutive powers of the changed primitive element are roots of transformed codewords. The correspondence

    α.sup.13 →(β.sup.7).sup.13 =β.sup.91.tbd.1 mod 15 =β                                                   (33)

as shown in TABLE III is used in a subsequent third transformation which yields g₁ (x).

The next transformation is a one-to-one mapping of codewords in Code B.2 onto codewords in Code B.3. The generator polynomial of Code B.3 is ##EQU38## Codeword polynomials C_(B).3 (x) must contain all the roots of g₂.3 (x) in order to contain g₂.3 (x) as a factor. Thus

    C.sub.B.3 (α.sup.13j)=C.sub.B.3 [(α.sup.7).sup.4j ]=0 for j=1, 2, . . . , 6                                              (35)

and C_(B).3 (x) is derived from C_(B).2 (x) by a permutation of the (symbol) coefficients of C_(B).2 (x). The permutation of the symbols of the latter is

    C.sub.[B.2]k →C.sub.[B.2]13k =C.sub.[B.3]k for k=0, 1, . . . , 14 (36)

where 13 k is reduced modulo 15. The permutation accounts for the substitution of α⁷ for α (resulting in a change in primitive element) to obtain g₂.3 (x) in Equation (34). Note that 13 (of 13 k in Equation (36) is the multiplicative inverse of 7 modulo 15.

Since (13,15)=1, 13 k modulo 15 is a permutation on the complete residue class k (0 1 . . . 14) modulo 15 as follows:

    __________________________________________________________________________     14                                                                               13                                                                               12                                                                               11                                                                               10                                                                                9                                                                                8                                                                               7 6 3  2                                                                                1                                                                               0 =  k of C.sub.k                                       2                                                                                4                                                                                6                                                                                8                                                                               10                                                                               12                                                                               14                                                                               1 7 9 11                                                                               13                                                                               0 =  13k mod 15                                        __________________________________________________________________________

EXAMPLE 12

Given codeword C_(B).2 from Example 11. The permutation C.sub.[B.2]13k yields codeword C_(B).3.

    __________________________________________________________________________     k of C.sub.k                                                                         14                                                                               13                                                                               12                                                                               11                                                                               10                                                                               9 8 7 6 5 4 3  2                                                                               1 0                                            __________________________________________________________________________     C.sub.B.2                                                                            14                                                                               * * * * * * * * 5 8 5 13                                                                               3 4                                            C.sub.B.3                                                                            *  3                                                                               * 13                                                                               * 5 * 8 * 5 * * 14                                                                               * 4                                            __________________________________________________________________________

Consider the coefficient C₁ =α³ of x in codeword polynomial C_(B).2 (x). Due to the permutation, it becomes the coefficient C₁₃ =α³ of x¹³ in codeword polynomial C_(B).3 (x). From Equation (34)

    α.sup.13j =(α.sup.7).sup.4j must be a root of C.sub.B.3 (x) for j=1, 2, . . . , 6

Evaluating the term α³ x¹³ in C_(B).3 (x) for

    x=α.sup.13 =(α.sup.7).sup.4 gives α.sup.3 [(α.sup.7).sup.4 ].sup.13 =α.sup.3 α.sup.4 =α.sup.7

It will now be shown that the permutation preserves the evaluation of the foregoing term in C_(B).2 (x) for the root (α⁷)⁴. From (31a)

    α.sup.4j must be a root of C.sub.B.2 (x) for j=1, 2, . . . , 6

Evaluating the term α³ x in C_(B).2 (x) for

    x=α.sup.4 gives α.sup.3 α.sup.4 =α.sup.7

as asserted.

Consider the coefficient C₃ =α⁵ of x³ in C_(B).2 (x). The evaluation of the term α⁵ x³ in C_(B).2 (x) for the root (α⁴)⁶ is

    α.sup.5 [(α.sup.4).sup.6 ].sup.3 =α.sup.5 [α.sup.12 ]=α.sup.2

The coefficient C₃ =α⁵ of x³ in C_(B).2 (x) becomes by permutation the coefficient C₃×13 mod 15 (=C₉) of x³×13 (=x⁹) in C_(B).3 (x). The evaluation of the term α⁵ x³×13 in C_(B).3 (x) for the root [(α⁷)⁴)]⁶ is

    α.sup.5 ([(α.sup.7).sup.4 ].sup.6).sup.3×13 =α.sup.5 [(α.sup.4).sup.6 ].sup.3 =α.sup.5 [α.sup.12 ]=α.sup.2

It may be further shown that

    C.sub.B.3 [(α.sup.7).sup.4j ]=C.sub.B.2 (α.sup.4j)=0 for j=1, 2, . . . , 6

where

    C.sub.13k mod 15 of C.sub.B.3 (x) equals C.sub.k of C.sub.B.2 (x) for k=0, 1, . . . , 14

The outcomes of successive transformations, thus far, on codeword (symbols) C_(B) in Example 10 are:

    __________________________________________________________________________     k of C.sub.k                                                                         14                                                                               13                                                                               12                                                                               11                                                                               10                                                                               9 8 7 6 5 4 3  2                                                                               1 0                                            __________________________________________________________________________     C.sub.B                                                                               7                                                                                0                                                                                0                                                                                0                                                                                0                                                                               0 0 0 0 7 A C  6                                                                               C A                                            C.sub.B.1                                                                            0 * * * * * * * * 0 4 2 11                                                                               2 4                                            C.sub.B.2                                                                            14                                                                               * * * * * * * * 5 8 5 13                                                                               3 4                                            C.sub.B.3                                                                            * 3 * 13                                                                               * 5 * 8 * 5 * * 14                                                                               * 4                                            __________________________________________________________________________

It may be verified that

    C.sub.B.3 (x)=(α.sup.3 x.sup.7 +α.sup.13 x.sup.6 +α.sup.12 x.sup.5 +α.sup.14 x.sup.4 +α.sup.13 x.sup.3 +α.sup.13 x.sup.2 +α.sup.10 x+α)g.sub.2.3 (x)

over GF(2⁴) defined by F_(B).3 (x) (=F_(B) (x)). The generator polynomial g₂.3 (x) is given in Equation (34). C_(B).3 in addition to C_(B).1 and C_(B).2 is associated with an encoder of the "conventional" type.

The fourth transformation is the one-to-one mapping of symbols in GF(2⁴) defined by F_(B) (x) onto symbols in GF(2⁴) defined by F_(G) (x). The mapping

    α.sup.i →β.sup.7i

where

    F.sub.B (α)=0 and F.sub.G (β)=0

is discussed above under the heading "Mathematical Characterization of RS Symbols" and appears in TABLE III. By mapping the RS symbols comprising a codeword in Code B.3 the corresponding codeword in Code G is obtained.

The generator polynomials g₂.3 (x) of C_(B).3 in (34) and g₁ (x) of C_(G) in Equation (19) reflect the foregoing mapping. ##EQU39##

As first introduced in (32) and (33) where the primitive element change was developed in preparation for the last transformation involving field element conversion

    α.sup.13j →(β.sup.7).sup.13j =βhu 91j.tbd.j mod 15=β.sup.j

The mapping of coefficients α^(i) of x^(j) to β^(7i) of x^(j) confirms the expanded form of g₁ (x). ##EQU40## Note that g₂.3 (x) and g₁ (x) are codeword polynomials of C_(B).3 (x) and C_(G) (x), respectively. The transformation (field element conversion) is applicable to every codeword polynomial in C_(B).3 (x). As discussed in the section on "Mathematical Characterization of RS Symbols," the mapping α^(i) →β⁷ is realizable by table look-up or by employing the linear transformation matrix ##EQU41##

EXAMPLE 13

Mapping the RS symbols of C_(B).3 in this example in GF(2⁴) defined by F_(B) (x) onto symbols in GF(2⁴) defined by F_(G) (x) results in codeword C_(G). Codeword C_(G) is the outcome of the fourth and last transformation in the following tabulation of successive transformations for mapping codeword C_(B) in Example 10 to C_(G).

    __________________________________________________________________________     k of C.sub.k                                                                         14                                                                               13                                                                               12                                                                               11                                                                               10                                                                               9 8  7                                                                               6 5 4 3  2                                                                               1 0                                            __________________________________________________________________________     C.sub.B                                                                               7                                                                                0                                                                                0                                                                                0                                                                                0                                                                               0 0  0                                                                               0 7 A C  6                                                                               C A                                            C.sub.B.1                                                                             0                                                                               * * * * * * * * 0 4 2 11                                                                               2 4                                            C.sub.B.2                                                                            14                                                                               * * * * * * * * 5 8 5 13                                                                               3 4                                            C.sub.B.3                                                                            * 3 * 13                                                                               * 5 *  8                                                                               * 5 * * 14                                                                               * 4                                            C.sub.G                                                                              * 6 *  1                                                                               * 5 * 11                                                                               * 5 * *  8                                                                               * 13                                           __________________________________________________________________________

The codeword polynomial C_(G) (x) contains g₁ (x) as a factor.

    C.sub.G (x)=(β.sup.6 x.sup.7 +βx.sup.6 +β.sup.9 x.sup.5 +β.sup.8 x.sup.4 +βx.sup.3 +β.sup.2 +β.sup.10 x+β.sup.7)g.sub.1 (x)

over GF(2⁴) defined by F_(G) (x).

The following are two examples of the results of a succession of transformations which map a given codeword in C_(B) onto one in C_(G). These represent typical test cases for testing the overall transformation because of the known structure of C_(B) or C_(G).

    __________________________________________________________________________     k of C.sub.k                                                                         14                                                                               13                                                                               12                                                                               11                                                                               10                                                                                9                                                                                8                                                                                7                                                                               6  5                                                                               4 3  2                                                                                1                                                                               0                                            __________________________________________________________________________     C.sub.B                                                                              D D D D D D D D D D D D D D D                                            C.sub.B.1                                                                             3                                                                                3                                                                                3                                                                                3                                                                                3                                                                                3                                                                                3                                                                                3                                                                               3  3                                                                               3 3  3                                                                                3                                                                               3                                            C.sub.B.2                                                                             2                                                                                1                                                                                0                                                                               14                                                                               13                                                                               12                                                                               11                                                                               10                                                                               9  8                                                                               7 6  5                                                                                4                                                                               3                                            C.sub.B.3                                                                            11                                                                                4                                                                               12                                                                                5                                                                               13                                                                                6                                                                               14                                                                                7                                                                               0  8                                                                               1 9  2                                                                               10                                                                               3                                            C.sub.G                                                                               2                                                                               13                                                                                9                                                                                5                                                                                1                                                                               12                                                                                8                                                                                4                                                                               0 11                                                                               7 3 14                                                                               10                                                                               6                                            __________________________________________________________________________

The 15 symbols of C_(G) above (expressed as i of β^(i)) are distinct.

    __________________________________________________________________________     k of C.sub.k                                                                         14                                                                               13                                                                               12                                                                               11                                                                               10                                                                               9 8  7                                                                               6  5                                                                                4                                                                               3 2 1 0                                            __________________________________________________________________________     C.sub.B                                                                              A A D  0                                                                                0                                                                               0 0  3                                                                               9  3                                                                                0                                                                               0 0  0                                                                               D                                            C.sub.B.1                                                                            4 4 3 * * * *  5                                                                               1  5                                                                               * * * * 3                                            C.sub.B.2                                                                            3 2 0 * * * * 12                                                                               7 10                                                                               * * * * 3                                            C.sub.B.3                                                                            * * * * * * * * 0 10                                                                                2                                                                               7 3 12                                                                               3                                            C.sub.G                                                                              * * * * * * * * 0 10                                                                               14                                                                               4 6  9                                                                               6                                            __________________________________________________________________________

The codeword polynomial C_(G) (x) with coefficients C_(G) above is g₁ (x), a codeword polynomial (of lowest degree).

Given two codewords in any (N,K) RS code. Every linear combination of the codewords is a codeword. For each of the four transformations discussed in connection with (15,9) RS codes, the transformation of the linear combination of two codewords is equal to the "sum" of the transformations on the two codewords. This is illustrated for the overall transformation of codewords C_(B) in Code B.1 to codewords C_(G) with intermediate results omitted.

EXAMPLE 14

Codeword C_(B) ³ (below) is a linear combination of codewords C_(B) ¹ and C_(B) ². Codewords C_(G) ³, C_(G) ¹ and C_(G) ² are the result of four successive transformations on C_(B) ³, C_(B) ¹ and C_(B) ² respectively.

    __________________________________________________________________________     k of Ck                                                                              14                                                                               13                                                                               12                                                                               11                                                                               10                                                                                9                                                                               8  7                                                                               6  5                                                                               4 3 2 1  0                                           __________________________________________________________________________     C.sub.B.sup.1                                                                         7                                                                                0                                                                               0 0 0  0                                                                               0  0                                                                               0  7                                                                               A C 6 C A                                            C.sub.B.sup.2                                                                         0                                                                                7                                                                               0 0 0  0                                                                               0  0                                                                               0 A F 5 B 9  4                                           C.sub.B.sup.3                                                                         7                                                                                7                                                                               0 0 0  0                                                                               0  0                                                                               0 D 5 9 5 E                                              C.sub.G.sup.1                                                                        *  6                                                                               * 1 *  5                                                                               * 11                                                                               *  5                                                                               * * 8 * 13                                           C.sub.G.sup.2                                                                        * 14                                                                               * 2 *  7                                                                               * 10                                                                               *  3                                                                               1 * * * 10                                           C.sub.G.sup.3                                                                        *  8                                                                               * 5 * 13                                                                               * 14                                                                               * 11                                                                               1 * 8 *  9                                           __________________________________________________________________________

Refer to Example 5 where RS symbols in GF(2⁴) defined by F_(G) (x) are tabulated to verify that the symbol-by-symbol "addition" of C_(G) ¹ and C_(G) ² gives C_(G) ³.

A received word R_(B) which may contain erroneous symbols may be viewed as a symbol-by-symbol "sum" of C_(B) with a symbol error pattern or sequence.

    R.sub.k =C.sub.k +E.sub.k for k=0, 1, . . . , 14

where R_(k) =C_(k) if and only if E_(k) is an all O's symbol.

Note that E_(k) is not to be confused with E, an integer which denotes the maximum number of erroneous RS symbols that are correctable. If the number of erroneous symbols is within the error correction capability R_(G) will be corrected by the decoder designed for Code G. Clearly R_(G) is the result of four successive transformations on the linear combination corresponding symbols comprising C_(B) and a symbol error sequence. The decoder determines the transformed error sequence and "subtracts" it from R_(G) to determine C_(G). The inverses of the successive transformations (which were applied to R_(B) to determine R_(G)) are successively applied to C_(G) in reverse order to recover C_(B). Clearly R_(B) (x), intermediate word polynomials and R_(G) (x) will not be divisible by their respective generator polynomials.

The inverses of transformations in order of their application on codewords C_(G) are summarized as follows:

1). Field element conversion inverse.

    β.sup.7i →α.sup.i or (β.sup.7).sup.13i =β.sup.i →α.sup.13i

Coefficients of codeword polynomials C_(G) (x) in GF(2⁴) defined by F_(G) (x) are mapped into corresponding coefficients of codeword polynomials C_(B).3 (x) in GF(2⁴) defined by F_(B) (x). See TABLE III. This is achieved by table look-up or employing the linear transformation matrix (as discussed in the section on "Mathematical Characterization of RS Symbols.") ##EQU42## and M.sub.βα is the inverse of M.sub.αβ associated with field element conversion.

2). Permutation Inverse

A primitive element change from α¹³ (resulting from field element conversion inverse β→α¹³) to α⁴ is required. The solution to

    (α.sup.w).sup.13 =α.sup.4 or 13w.tbd.4 mod 15

is w=13 the inverse of y=7 in Equation (32). The respective generator polynomials for Codes G, B.3, B.2, B.1 and B are fixed. In the reverse applications of inverse transformations, codeword polynomials C_(B).2 (x) must contain all the roots of g₂.2 (x) in order to contain g₂.2 (x) as a factor. Thus

    C.sub.B.2 (α.sup.4j)=C.sub.B.2 [(α.sup.13).sup.13j ]=0 for j=1, 2, . . . , 6

and C_(B).2 (x) is derived from C_(B).3 (x) by the following permutation of the coefficients of C_(B).3 (x).

    C.sub.[B.3]k →C.sub.[B.3]7k =C.sub.[B.2]k for k=0, 1, . . . , 14

where 7 k is reduced modulo 15. The foregoing arguments follow those that resulted in expressions (34), (35) and (36). The permutation 13 k in (36) subsequently followed by its inverse

    k→13k→13(7)k mod 15=k

returns the symbols of C_(B).2 (i.e., symbols of R_(B).2 corrected) to their original position.

3). Translation of the Powers of Roots (of g₂.2 (x)) Inverse

Of interest is the effect of the inverse translation of the powers of roots of g₂.2 (x). Given ##EQU43## and C_(B).2 (α¹⁴ x) is a Code B.1 codeword polynomial. 4). Conversion from {α^(i) } Basis to {l_(j) } Basis

See TABLE IV. This can be done by table look-up or by using the linear transformation matrix ##EQU44## T.sub.αl is the inverse of ##EQU45##

EXAMPLE 15

R_(B) is a received word originating from Code B with erroneous symbols. Successive transformations are applied to obtain R_(G). R_(G) is decoded whereby erroneous symbols are corrected to determine C_(G) a valid codeword in Code G. Successive inverse transformations are then applied to C_(G) to recover C_(B), the codeword in Code B most likely to have been sent.

Symbols R₁₄, R₅ and R₂ in word R_(B) are in error. Since three erroneous symbols are within the symbol error-correcting capability (i.e., 2E=15-9 and E=3), they are corrected when R_(G) is decoded into C_(G). Italicized entries in the following table are symbol changes resulting from the decoding process.

    __________________________________________________________________________     k of C.sub.k                                                                          14                                                                               13                                                                               12                                                                               11                                                                               10                                                                                9                                                                               8  7                                                                                6                                                                                5                                                                                4                                                                                3                                                                                2                                                                                1                                                                                0                                          __________________________________________________________________________     R.sub.B                                                                               D  3                                                                               F  6                                                                                9                                                                               D 2 E  5                                                                                7                                                                                2                                                                               C  9                                                                                2                                                                               F                                           R.sub.B.1                                                                             3  5                                                                                6                                                                               11                                                                                1                                                                                3                                                                               7 12                                                                               13                                                                                0                                                                                7                                                                                2                                                                                1                                                                                7                                                                                6                                          R.sub.B.2                                                                             2  3                                                                                3                                                                                7                                                                               11                                                                               12                                                                               0  4                                                                                4                                                                                5                                                                               11                                                                                5                                                                                3                                                                                8                                                                                6                                          R.sub.B.3                                                                             0  8                                                                               12                                                                                3                                                                               11                                                                                5                                                                               7 11                                                                                3                                                                                5                                                                                3                                                                                4                                                                                2                                                                                4                                                                                6                                          R.sub.G                                                                               0 11                                                                                9                                                                                6                                                                                2                                                                                5                                                                               4  2                                                                                6                                                                                5                                                                                6                                                                               13                                                                               14                                                                               13                                                                               12                                          C.sub.G                                                                               0 11                                                                                9                                                                                2                                                                                2                                                                                5                                                                               4  2                                                                                6                                                                               13                                                                                6                                                                               13                                                                                 6                                                                              13                                                                               12                                          C.sub.B.3                                                                             0  8                                                                               12                                                                               11                                                                               11                                                                                5                                                                               7 11                                                                                3                                                                                4                                                                                3                                                                                4                                                                                3                                                                                4                                                                                6                                          C.sub.B.2                                                                             3  3                                                                                3                                                                                7                                                                               11                                                                               12                                                                               0  4                                                                                4                                                                                4                                                                               11                                                                                5                                                                               11                                                                                8                                                                                6                                          C.sub.B.1                                                                             4  5                                                                                6                                                                               11                                                                                1                                                                                3                                                                               7 12                                                                               13                                                                               14                                                                                7                                                                                2                                                                                9                                                                                7                                                                                6                                          C.sub.B                                                                               A  3                                                                               F  6                                                                                9                                                                               D 2 E  5                                                                                1                                                                                2                                                                               C B  2                                                                               F                                           __________________________________________________________________________

The Transformational Equivalence of Berlekamp and Conventional RS Codes for (N,K)=(255,223)

A hardware ground decoder has been designed and built and is operating to decode a (255,223) RS code. The code is associated with a conventional encoder aboard the in-flight interplanetary Galileo and Voyager spacecraft. All future interplanetary space probes, starting with Mars Observer, that utilize RS encoding must use the Berlekamp representation detailed in "Telemetry Channel Coding" Recommendation, CCSDS 101.0-B-2 Blue Book, Consultive Committee for Space Data Systems, January 1987. A succession of transformations and their inverses have been developed, programmed and incorporated into the system hardware. Words originating from Code B are mapped (by means of successive transformations) onto words in Code G, and decoded for symbol error correction. If the number of erroneous symbols are within the error correction capability of the code, the decoded word is a codeword in Code G. It is then mapped (by means of successive inverse transformations applied in reverse order) onto the codeword in Code B most likely to have been transmitted. The parameters of the codes are J=8 and E=16.

FIGS. 4a, b, c and d illustrate a flow chart of the process for transforming words R_(B) with possible errors in a Berlekamp code to words R_(G) in a conventional code for decoding the words R_(G) having possible errors to codewords C_(G) free of error. That is accomplished in blocks 100, 110, 120, 130, 140 and 150 in FIGS. 4a and b. The corrected codewords C_(G) are then reverse transformed symbol-by-symbol back into codewords C_(B) in the Berlekamp code. The uncorrected words in the Berlekamp code and the conventional code are here denoted by the letter R with subscripts B and G, respectively, because they are with possible error. Until corrected in block 150, FIG. 4b, they are not denoted codewords by the letter C with first the subscript G and later in the process the subscript B. The codewords C_(G) are not information, because in the process of transformation from R_(B) to R_(G) for decoding (i.e., correcting errors) they have undergone permutation. To retrieve the information, the codewords C_(G) must be transformed back into codewords C_(B) by inverse transformation in a reverse order of process steps, as will be illustrated by the following examples.

A mathematical description of each of the (255,223) RS codes is presented as follows: ##EQU46## Elements in GF(2⁸) defined by F_(B) (x) in Equation (37) represented in basis {α^(i) } and basis {l_(j) } appear in TABLE V.

                  TABLE V                                                          ______________________________________                                         Elements of GF(2.sup.8) Defined by F.sub.B (x) = x.sup.8 +                     x.sup.7 + x.sup.2 + x + 1 in Basis {α.sup.i } and Basis {l.sub.j         }.                                                                             i of α.sup.i                                                                        i of b.sub.i                                                                            Tr(α.sup.i)                                                                          j of v.sub.j                                   ______________________________________                                                    76543210             01234567                                       *          00000000 0           00000000                                       0          00000001 0           01111011                                       1          00000010 1           10101111                                       2          00000100 1           10011001                                       3          00001000 1           11111010                                       4          00010000 1           10000110                                       5          00100000 1           11101100                                       6          01000000 1           11101111                                       7          10000000 1           10001101                                       8          10000111 1           11000000                                       9          10001001 0           00001100                                       10         10010101 1           11101001                                       11         10101101 0           01111001                                       12         11011101 1           11111100                                       13         00111101 0           01110010                                       14         01111010 1           11010000                                       15         11110100 1           10010001                                       16         01101111 1           10110100                                       17         11011110 0           00101000                                       18         00111011 0           01000100                                       19         01110110 1           10110011                                       20         11101100 1           11101101                                       21         01011111 1           11011110                                       22         10111110 0           00101011                                       23         11111011 0           00100110                                       24         01110001 1           11111110                                       25         11100010 0           00100001                                       26         01000011 0           00111011                                       27         10000110 1           10111011                                       28         10001011 1           10100011                                       29         10010001 0           01110000                                       30         10100101 1           10000011                                       31         11001101 0           01111010                                       32         00011101 1           10011110                                       33         00111010 0           00111111                                       34         01110100 0           00011100                                       35         11101000 0           01110100                                       36         01010111 0           00100100                                       37         10101110 1           10101101                                       38         11011011 1           11001010                                       39         00110001 0           00010001                                       40         01100010 1           10101100                                       41         11000100 1           11111011                                       42         00001111 1           10110111                                       43         00011110 0           01001010                                       44         00111100 0           00001001                                       45         01111000 0           01111111                                       46         11110000 0           00001000(l.sub.4)                              47         01100111 0           01001110                                       48         11001110 1           10101110                                       49         00011011 1           10101000                                       50         00110110 0           01011100                                       51         01101100 0           01100000                                       52         11011000 0           00011110                                       53         00110111 0           00100111                                       54         01101110 1           11001111                                       55         11011100 1           10000111                                       56         00111111 1           11011101                                       57         01111110 0           01001001                                       58         11111100 0           01101011                                       59         01111111 0           00110010                                       60         11111110 1           11000100                                       61         01111011 1           10101011                                       62         11110110 0           00111110                                       63         01101011 0           00101101                                       64         11010110 1           11010010                                       65         00101011 1           11000010                                       66         01010110 0           01011111                                       67         10101100 0           00000010(l.sub.6)                              68         11011111 0           01010011                                       69         00111001 1           11101011                                       70         01110010 0           00101010                                       71         11100100 0           00010111                                       72         01001111 0           01011000                                       73         10011110 1           11000111                                       74         10111011 1           11001001                                       75         11110001 0           01110011                                       76         01100101 1           11100001                                       77         11001010 0           00110111                                       78         00010011 0           01010010                                       79         00100110 1           11011010                                       80         01001100 1           10001100                                       81         10011000 1           11110001                                       82         10110111 1           10101010                                       83         11101001 0           00001111                                       84         01010101 1           10001011                                       85         10101010 0           00110100                                       86         11010011 0           00110000                                       87         00100001 1           10010111                                       88         01000010 0           01000000(l.sub.1)                              89         10000100 0           00010100                                       90         10001111 0           00111010                                       91         10011001 1           10001010                                       92         10110101 0           00000101                                       93         11101101 1           10010110                                       94         01011101 0           01110001                                       95         10111010 1           10110010                                       96         11110011 1           11011100                                       97         01100001 0           01111000                                       98         11000010 1           11001101                                       99         00000011 1           11010100                                       100        00000110 0           00110110                                       101        00001100 0           01100011                                       102        00011000 0           01111100                                       103        00110000 0           01101010                                       104        01100000 0           00000011                                       105        11000000 0           01100010                                       106        00000111 0           01001101                                       107        00001110 1           11001100                                       108        00011100 1           11100101                                       109        00111000 1           10010000                                       110        01110000 1           10000101                                       111        11100000 1           10001110                                       112        01000111 1           10100010                                       113        10001110 0           01000001                                       114        10011011 0           00100101                                       115        10110001 1           10011100                                       116        11100101 0           01101100                                       117        01001101 1           11110111                                       118        1011010  0           01011110                                       119        10110011 0           00110011                                       120        11100001 1           11110101                                       121        01000101 0           00001101                                       122        10001010 1           11011000                                       123        10010011 1           11011111                                       124        10100001 0           00011010                                       125        11000101 1           10000000(l.sub.0)                              126        00001101 0           00011000                                       127        00011010 1           11010011                                       128        00110100 1           11110011                                       129        01101000 1           11111001                                       130        11010000 1           11100100                                       131        00100111 1           10100001                                       132        01001110 0           00100011                                       133        10011100 0           01101000                                       134        10111111 0           01010000                                       135        11111001 1           10001001                                       136        01110101 0           01100111                                       137        11101010 1           11011011                                       138        01010011 1           10111101                                       139        10100110 0           01010111                                       140        11001011 0           01001100                                       141        00010001 1           11111101                                       142        00100010 0           01000011                                       143        01000100 0           01110110                                       144        10001000 0           01110111                                       145        10010111 0           01000110                                       146        10101001 1           11100000                                       147        11010101 0           00000110                                       148        00101101 1           11110100                                       149        01011010 0           00111100                                       150        10110100 0           01111110                                       151        11101111 0           00111001                                       152        01011001 1           11101000                                       153        10110010 0           01001000                                       154        11100011 0           01011010                                       155        01000001 1           10010100                                       156        10000010 0           00100010                                       157        10000011 0           01011001                                       158        10000001 1           11110110                                       159        10000101 0           01101111                                       160        10001101 1           10010101                                       161        10011101 0           00010011                                       162        10111101 1           11111111                                       163        11111101 0           00010000(l.sub.3)                              164        01111101 1           10011101                                       165        11111010 0           01011101                                       166        01110011 0           01010001                                       167        11100110 1           10111000                                       168        01001011 1           11000001                                       169        10010110 0           00111101                                       170        10101011 0           01001111                                       171        11010001 1           10011111                                       172        00100101 0           00001110                                       173        01001010 1           10111010                                       174        10010100 1           10010010                                       175        10101111 1           11010110                                       176        11011001 0           01100101                                       177        00110101 1           10001000                                       178        01101010 0           01010110                                       179        11010100 0           01111101                                       180        00101111 0           01011011                                       181        01011110 1           10100101                                       182        10111100 1           10000100                                       183        iiiiiiii 1           10111111                                       184        01111001 0           00000100(l.sub.5)                              185        11110010 1           10100111                                       186        01100011 1           11010111                                       187        11000110 0           01010100                                       188        00001011 0           00101110                                       189        00010110 1           10110000                                       190        00101100 1           10001111                                       191        01011000 1           10010011                                       192        10110000 1           11100111                                       193        11100111 1           11000011                                       194        01001001 0           01101110                                       195        10010010 1           10100100                                       196        10100011 1           10110101                                       197        11000001 0           00011001                                       198        00000101 1           11100010                                       199        00001010 0           01010101                                       200        00010100 0           00011111                                       201        00101000 0           00010110                                       202        01010000 0           01101001                                       203        10100000 0           01100001                                       204        11000111 0           00101111                                       205        00001001 I           10000001                                       206        00010010 0           00101001                                       207        00100100 0           01110101                                       208        01001000 0           00010101                                       209        10010000 0           00001011                                       210        10100111 0           00101100                                       211        11001001 1           11100011                                       212        00010101 0           01010100                                       213        00101010 1           10111001                                       214        01010100 1           11110000                                       215        10101000 1           10011011                                       216        11010111 1           10101001                                       217        00101001 0           01101101                                       218        01010010 1           11000110                                       219        10100100 1           11111000                                       220        11001111 1           11010101                                       221        00011001 0           00000111                                       222        00110010 1           11000101                                       223        01100100 1           10011010                                       224        11001000 1           10011000                                       225        00010111 1           11001011                                       226        00101110 0           00100000(l.sub.2)                              227        01011100 0           00001010                                       228        10111000 0           00011101                                       2˜9  11110111 0           01000101                                       230        01101001 1           10000010                                       231        11010010 0           01001011                                       232        00100011 0           00111000                                       233        01000110 1           11011001                                       234        10001100 1           11101110                                       235        10011111 1           10111100                                       236        10111001 0           01100110                                       237        11110101 1           11101010                                       238        01101101 0           00011011                                       239        11011010 1           10110001                                       240        00110011 1           10111110                                       241        01100110 0           00110101                                       242        11001100 0           00000001(l.sub.7)                              243        00011111 0           00110001                                       244        00111110 1           10100110                                       245        01111100 1           11100110                                       246        11111000 1           11110010                                       247        01110111 1           11001000                                       248        11101110 0           01000010                                       249        01011011 0           01000111                                       250        10110110 1           11010001                                       251        11101011 1           10100000                                       252        01010001 0           00010010                                       253        10100010 1           11001110                                       254        11000011 1           10110110                                       ______________________________________                                    

The coefficients (G₃₂, G₃₁, . . . , G₀) of g₂ (x) in Equation (38), the self-reciprocal generator polynomial over GF(2⁸) of Code B are

    __________________________________________________________________________     G.sub.32 →                                                                  0  249                                                                               59                                                                               66 4  43                                                                               126                                                                               251                                                                               97                                                                               30 3                                                   213                                                                               50 66                                                                               170                                                                               5  24                                                                               5  170                                                                               66                                                                               50 213                                                 3  30 97                                                                               251                                                                               126                                                                               43                                                                               4  66 59                                                                               249                                                                               0  ← G.sub.0                               __________________________________________________________________________

expressed as i of α^(i) in TABLE V. Note that are 15 distinct nonzero entries and

    G.sub.i =G.sub.32-i and G.sub.3 =G.sub.29 =G.sub.13 =G.sub.19 =66 (i.e., α.sup.66)

(The {T_(l) } linear functions of z_(i) components for designing the linear binary matrix in FIG. 2 can be found in Ref. [3]).

2. Code G

    F.sub.G (x)=x.sup.8 +x.sup.4 +x.sup.3 +x.sup.2 +1 defines GF(2.sup.8)

where β⁸ =β⁴ +β³ +β² +1 ##EQU47## Elements in GF(2⁸) defined by F_(G) (x) in (41) are given in TABLE VI. The coefficients (G₃₂, G₃₁, . . . , G₀) of g₁ (x) in (41), the generator polynomial over GF(2⁸) of Code G are

    __________________________________________________________________________     G.sub.32 →                                                                  0  11 8 109                                                                               194                                                                               254                                                                               173                                                                               11 75 218                                                                               148                                               149                                                                               44 0 137                                                                               104                                                                               43 137                                                                               203                                                                               99 176                                                                               59                                                91 194                                                                               84                                                                               53 248                                                                               107                                                                               80 28 215                                                                               251                                                                               18 ← G.sub.0                             __________________________________________________________________________

expressed as i of β^(i) in TABLE VI. Note that there 28 distinct nonzero entries corresponding to 28 different multipliers.

Transformations between Berlekamp and Galileo (Voyager) (255,223) RS codes are summarized as follows:

1). Conversion from {l_(j) } Basis and {α^(i) } Basis (Block 110) and its Inverse (Block 190).

Refer to TABLE V. Symbol-by-symbol conversion may be provided by table look-up. Also post-multiplication on the 8-bit binary vector representation of a symbol by the linear transformation matrix T_(l)α realizes the conversion. The transformation matrix and its inverse is derived from TABLE V.

                  TABLE VI                                                         ______________________________________                                         Elements in GF(2.sup.8) Defined by                                             F.sub.G (x) = x.sup.8 + x.sup.4 + x.sup.3 + x.sup.2 + 1.                              i of β.sup.i                                                                    i of c.sub.i                                                      ______________________________________                                                      76543210                                                                 *     00000000                                                                 0     00000001                                                                 1     00000010                                                                 2     00000100                                                                 3     00001000                                                                 4     00010000                                                                 5     00100000                                                                 6     01000000                                                                 7     10000000                                                                 8     00011101                                                                 9     00111010                                                                 10    01110100                                                                 11    11101000                                                                 12    11001101                                                                 13    10000111                                                                 14    00010011                                                                 15    00100110                                                                 16    01001100                                                                 17    10011000                                                                 18    00101101                                                                 19    01011010                                                                 20    10110100                                                                 21    01110101                                                                 22    11101010                                                                 23    11001001                                                                 24    10001111                                                                 25    00000011                                                                 26    00000110                                                                 27    00001100                                                                 28    00011000                                                                 29    00110000                                                                 30    01100000                                                                 31    11000000                                                                 32    10011101                                                                 33    00100111                                                                 34    01001110                                                                 35    10011100                                                                 36    00100101                                                                 37    01001010                                                                 38    10010100                                                                 39    00110101                                                                 40    01101010                                                                 41    11010100                                                                 42    10110101                                                                 43    01110111                                                                 44    11101110                                                                 45    11000001                                                                 46    10011111                                                                 47    00100011                                                                 48    01000110                                                                 49    10001100                                                                 50    00000101                                                                 51    00001010                                                                 52    00010100                                                                 53    00101000                                                                 54    01010000                                                                 55    10100000                                                                 56    01011101                                                                 57    10111010                                                                 58    01101001                                                                 59    11010010                                                                 60    10111001                                                                 61    01101111                                                                 62    11011110                                                                 63    10100001                                                                 64    01011111                                                                 65    10111110                                                                 66    01100001                                                                 67    11000010                                                                 68    10011001                                                                 69    00101111                                                                 70    01011110                                                                 71    10111100                                                                 72    01100101                                                                 73    11001010                                                                 74    10001001                                                                 75    00001111                                                                 76    00011110                                                                 77    00111100                                                                 78    01111000                                                                 79    11110000                                                                 80    11111101                                                                 81    11100111                                                                 82    11010011                                                                 83    10111011                                                                 84    01101011                                                                 85    11010110                                                                 86    10110001                                                                 87    01111111                                                                 88    11111110                                                                 89    11100001                                                                 90    11011111                                                                 91    10100011                                                                 92    01011011                                                                 93    10110110                                                                 94    01110001                                                                 95    11100010                                                                 96    11011001                                                                 97    10101111                                                                 98    01000011                                                                 99    10000110                                                                 100   00010001                                                                 101   00100010                                                                 102   01000100                                                                 103   10001000                                                                 104   00001101                                                                 105   00011010                                                                 106   00110100                                                                 107   01101000                                                                 108   11010000                                                                 109   10111101                                                                 110   01100111                                                                 111   11001110                                                                 112   10000001                                                                 113   00011111                                                                 114   00111110                                                                 115   01111100                                                                 116   11111000                                                                 117   11101101                                                                 118   11000111                                                                 119   10010011                                                                 120   00111011                                                                 121   01110110                                                                 122   11101100                                                                 123   11000101                                                                 124   10010111                                                                 125   00110011                                                                 126   01100110                                                                 127   11001100                                                                 128   10000101                                                                 129   00010111                                                                 130   00101110                                                                 131   01011100                                                                 132   10111000                                                                 133   01101101                                                                 134   11011010                                                                 135   10101001                                                                 136   01001111                                                                 137   10011110                                                                 138   00100001                                                                 139   01000010                                                                 140   10000100                                                                 141   00010101                                                                 142   00101010                                                                 143   01010100                                                                 144   10101000                                                                 145   01001101                                                                 146   10011010                                                                 147   00101001                                                                 148   01010010                                                                 149   10100100                                                                 150   01010101                                                                 151   10101010                                                                 152   01001001                                                                 153   10010010                                                                 154   00111001                                                                 155   01110010                                                                 156   11100100                                                                 157   11010101                                                                 158   10110111                                                                 159   01110011                                                                 160   11100110                                                                 161   11010001                                                                 162   10111111                                                                 163   01100011                                                                 164   11000110                                                                 165   10010001                                                                 166   00111111                                                                 167   01111110                                                                 168   11111100                                                                 169   11100101                                                                 170   11010111                                                                 171   10110011                                                                 172   01111011                                                                 173   11110110                                                                 174   11110001                                                                 175   11111111                                                                 176   11100011                                                                 177   11011011                                                                 178   10101011                                                                 179   01001011                                                                 180   10010110                                                                 181   00110001                                                                 182   01100010                                                                 183   11000100                                                                 184   10010101                                                                 185   00110111                                                                 186   01101110                                                                 187   11011100                                                                 188   10100101                                                                 189   01010111                                                                 190   10101110                                                                 191   01000001                                                                 192   10000010                                                                 193   00011001                                                                 194   00110010                                                                 195   01100100                                                                 196   11001000                                                                 197   10001101                                                                 198   00000111                                                                 199   00001110                                                                 200   00011100                                                                 201   00111000                                                                 202   01110000                                                                 203   11100000                                                                 204   11011101                                                                 205   10100111                                                                 206   01010011                                                                 207   10100110                                                                 208   01010001                                                                 209   10100010                                                                 210   01011001                                                                 211   10110010                                                                 212   01111001                                                                 213   11110010                                                                 214   11111001                                                                 215   11101111                                                                 216   11000011                                                                 217   10011011                                                                 218   00101011                                                                 219   01010110                                                                 220   10101100                                                                 221   01000101                                                                 222   10001010                                                                 223   00001001                                                                 224   00010010                                                                 225   00100100                                                                 226   01001000                                                                 227   10010000                                                                 228   00111101                                                                 229   01111010                                                                 230   11110100                                                                 231   11110101                                                                 232   11110111                                                                 233   11110011                                                                 234   11111011                                                                 235   11101011                                                                 236   11001011                                                                 237   10001011                                                                 238   00001011                                                                 239   00010110                                                                 240   00101100                                                                 241   01011000                                                                 242   10110000                                                                 243   01111101                                                                 244   11111010                                                                 245   11101001                                                                 246   11001111                                                                 247   10000011                                                                 248   00011011                                                                 249   00110110                                                                 250   01101100                                                                 251   11011000                                                                 252   10101101                                                                 253   01000111                                                                 254   10001110                                                          ______________________________________                                          ##EQU48##

Table look-up can provide the inverse transformation or ##EQU49## 2). Translation of Powers of Roots in g₂.1 (x) (Block 120) and its Inverse (Block 180).

The degree 32 generator polynomial g₂.1 (x)=g₂ (x) as given in (38). Translating j running from 112 to 143 in the product summation form to run from 1 to 32 requires a change in the argument x. See Ref. [12] and the discussion in Section 5.A. (dealing with (15,9) RS codes) leading to Example 11 and subsequent expressions involving the effect of the inverse translation. Omitting the step-by-step derivation of g₂.2 (x) from g₂.1 (x) and codeword polynomial C_(B).2 (x) from C_(B).1 (x) the results over GF(2⁸) defined by F_(B) (x) are ##EQU50## The coefficients (G₃₂, G₃₁, . . . , G₀) of g₂.2 (x) in (42), the generator polynomial over GF(2⁸) of Code B.2 are T2 -G₃₂ →? 0 48 167 228 220 58 195 119 19 63 3 ? - 42 188 3 161 50 123 158 122 72 110 72 - 171 252 118 71 0 226 241 102 149 138 198 ← G_(0?) -

expressed as i of α^(i) in TABLE V.

Following are the results associated with the inverse translation: ##EQU51## 3). Permutation (Block 130) and its Inverse (Block 170) which Accounts for a Primitive Element Change

In preparation for converting field elements defined by F_(B) (x) to those defined by F_(G) (x) an isomorphism between the two representations of GF(2⁸) must be established. See Section 2. and Ref. [2]. It may be verified from TABLES V and VI that the one-to-one mapping

    α.sup.i ⃡β.sup.-43i =β.sup.212i

is an isomorphism. The elements α and β²¹² have the same minimal polynomial, namely, F_(B) (x). Also

    α.sup.83 →(β.sup.212).sup.83 =β since 212×83.tbd.1 mod 255

A primitive element change in (42) from α¹¹ to α⁸³ is needed prior to field element conversion. The unique solution to

    (α.sup.y).sup.11 =α.sup.83 or 11y.tbd.83 mod 255

is

    y=193

and the inverse of y is 37 (i.e., 193×37.tbd.1). The primitive element change in (42) corresponds to g₂.2 (x) becoming ##EQU52## In order for C_(B).3 (x) to contain g₂.3 (x) as a factor, the coefficients of C_(B).3 (x) is a permutation of the coefficients of C_(B).2 (x). That is

    C.sub.[B.2]k →C.sub.[B.2]37k =C.sub.[B.3]k for k=0, 1, . . . , 254 (47)

Since 37 (the inverse of 193) has no common factor with 255 (=3×5×17) the permutation in (47) is guaranteed (i.e., no two coefficients will map to the same position).

Similarly the inverse permutation is associated with the change of primitive element from

    α.sup.83 to (α.sup.37).sup.83 =α.sup.11

The coefficients of C_(B).2 (x) is the following permutation of the coefficients of C_(B).2 (x)

    C.sub.[B.3]k →C.sub.[B.3]193k =C.sub.[B.2]k for k=0, 1, . . . , 254 (48)

The coefficients (G₃₂, G₃₁, . . . , G₀) of g₂.3 (x) in (46), the generator polynomial over GF(2⁸) of Code B.3 are

    __________________________________________________________________________     G.sub.32 →                                                                  0  148                                                                               154                                                                               122                                                                               37 172                                                                               79 148                                                                               105                                                                               244                                                                               44                                               127                                                                               82 0  151                                                                               217                                                                               254                                                                               151                                                                               19 57 73 52                                               158                                                                               37 87 64 184                                                                               211                                                                               10 29 250                                                                               178                                                                               219                                                                               ← G.sub.0                            __________________________________________________________________________

expressed as i of α^(i) in TABLE V.

4). Field Element Conversion (Block 140) and its Inverse (Block 160).

Coefficients of codeword polynomials C_(B).3 (x) (including g₂.3 (x), a codeword polynomial of minimum degree) in GF(2⁸) defined by F_(B) (x) are mapped into corresponding coefficients of codeword polynomials C_(G) (x). The one-to-one mapping

    α.sup.i →β.sup.212i

is achievable by table look-up. The conversion can also be done by the linear matrix equation ##EQU53##

The inverse field element transformation is the symbol-by-symbol mapping from GF(2⁸) defined by F_(G) (x) to GF(2⁸) defined by F_(B) (x) the symbols to be mapped are decoded word symbols emanating from the ground decoder. The one-to-one mapping is

    β.sup.212i →α.sup.i or (β.sup.212i).sup.83 →(α.sup.i).sup.83 =β.sup.i →α.sup.83i

Mapping can be done by table look-up or by means of the linear matrix equation ##EQU54## It should be again noted that because of the linearity of the RS codes and the transformations the symbols of a word originating from Code B are

    R.sub.k =C.sub.k +E.sub.k for k=0, 1, . . . , 254

and R_(k) is a linear sum is transformed until the word R_(G) is obtained. Upon decoding R_(G), C_(G) is obtained if the number of erroneous symbols is within the error correction capability of the (255,223) RS code. See Example 15. Symbols in error remain in error after successive forward transformations even if the number of erroneous symbols exceed the error correction capability of the RS code.

Although specific examples have been given of the transformation of the (Berlekamp) code, C_(B), to a specific conventional code, C_(G), it should be appreciated that the invention is useful in transforming any (N,K) RS code over GF(2^(J)) to another (N,K) RS code over GF(2^(J)). This RS coding for error protection has wide applications in communication channels (from here to there) and storage devices (from now until then), for example, magnetic disc or tape, optical disc, and solid-stage memories. 

I claim:
 1. A process for realizing mappings between codewords of two distinct (N,K) Reed-Solomon codes over GF(2^(J)) having selected two independent parameters: J, specifying the number of bits per symbol; and E, the symbol error correction capability of the code, wherein said independent parameters J and E yield the following: N=2^(J) -1, total number of symbols per codeword; 2E, the number of symbols assigned a role of check symbols; and K=N-2E, the number of code symbols representing information, all within a codeword of an (N,K) RS code over GF(2^(J)), and having selected said parameters for encoding, the implementation of a decoder are governed by: 2^(J) field elements defined by a degree J primitive polynomial over GF(2) denoted by F(x); a code generator polynomial of degree 2E containing 2E consecutive roots of a primitive element defined by F(x); and, in a Berlekamp RS code, the basis in which the RS information and check symbols are represented, said process comprising the sequential steps of:first, symbol-by-symbol conversion due to root power translation of words R_(B).1 of a first RS code B to words R_(B).2, second, permutation of symbols of words R_(B).2 to yield words R_(B).3, third, symbol-by-symbol conversion of words R_(B).3 to words R_(G) of a second RS code G, fourth, decoding words R_(G) by correcting erroneous symbols in words R_(G) using check symbols of each of said words R_(G) for the information symbols in each word R_(G) to obtain corrected words C_(G), and fifth, transformation of said corrected words C_(G) back to said code B by the inverse of said third, second and first steps in that reverse sequence to convert codewords C_(G) to codeword C_(B).3, to permute codewords C_(B).3 to yield codewords C_(B).2, and symbol-by-symbol conversion due to root power translation to convert codewords C_(B).2 to codewords C_(B).1.
 2. A process as defined in claim 1 wherein said words R_(B).1 of said first code B are defined by a function F(x) wherein said RS symbols are represented in another representation using a parameter λ for a field element where 1, λ, λ², . . . , λ^(r-1) is a basis in GF(2^(r)), and said process includes an initial step of converting symbol-to-symbol words C_(B) of a first basis to said words R_(B) of a second basis and a final step of said reverse sequence of steps for symbol-to-symbol conversion from said second basis to said first basis of representation using said parameter λ. 